124
Baryogenesis
where W/lII ' B/lII are the usual field strengths, Dit/J is the gauge covariant
derivative
Dit/J = ait/J + ig2 Wit/J + iig) Bit/J
(4.182)
and
V(t/J) = }..(t/Jtt/J - !v 2 )2.
(4.183)
In the first instance, the U (1) fields are ignored, i.e. Ow = O. Substituting the
ansatz gives the energy in the fonn
E = 2mw (00 d~ F (~, I, I',h. h'; mH)
(4.184)
a2 Jo
mw
where ~ = mwr, so that I and h are functions of ~ alone, and m~ = 2}..v 2 is the
Higgs mass. The Euler-Lagrange equations which follow from requiring E to be
minimized may be solved approximately or numerically, with I and h required
to approach zero as ~ ~ 00. Substituting the solutions back into E gives the
minimum energy in the fonn
Eroin = 2m w E (m H) .
(4.185)
a2
mw
The scale is set by the prefactor
2mw :::::: 5TeV
(4.186)
a2
and the function e is rather slowly varying: it increases only from 1.5 to 2.7 as
m H increases from zero to infinity, taking the value 2.1 when m H = 2.8m w. In
the range 25 Ge V < m H < 250 Ge V, E is well approximated by
E(x) = 1.58 + 0.32x - O.05x 2 •
(4.187)
Allowing Ow ::f: 0 changes Emin by about 100 GeV.
Although the configuration has the minimum energy among those satisfying
the ansatz, it is nevertheless unstable against perturbations which do not satisfy
the ansatz. This was to be expected too from the schematic diagram in figure 4.9,
in which the top of the potential barrier is evidently a maximum. Because of this
instability, the configuration was named a 'sphaleron', from the Greek meaning
'ready to fall'. The energy of the sphaleron satisfies
8 TeV < Espb < 14 TeV
(4.188)
and measures the height of the saddle point in configuration space over which
the vacuum must be 'pushed' to reach a topologically distinct vacuum. This
suggests that the exponentially small tunnelling rate (4.174) might be evaded by
supplying 0( I 0 Te V) of energy, for example in a pn collision. Then some baryon
Baryogenesis
where W/lII ' B/lII are the usual field strengths, Dit/J is the gauge covariant
derivative
Dit/J = ait/J + ig2 Wit/J + iig) Bit/J
(4.182)
and
V(t/J) = }..(t/Jtt/J - !v 2 )2.
(4.183)
In the first instance, the U (1) fields are ignored, i.e. Ow = O. Substituting the
ansatz gives the energy in the fonn
E = 2mw (00 d~ F (~, I, I',h. h'; mH)
(4.184)
a2 Jo
mw
where ~ = mwr, so that I and h are functions of ~ alone, and m~ = 2}..v 2 is the
Higgs mass. The Euler-Lagrange equations which follow from requiring E to be
minimized may be solved approximately or numerically, with I and h required
to approach zero as ~ ~ 00. Substituting the solutions back into E gives the
minimum energy in the fonn
Eroin = 2m w E (m H) .
(4.185)
a2
mw
The scale is set by the prefactor
2mw :::::: 5TeV
(4.186)
a2
and the function e is rather slowly varying: it increases only from 1.5 to 2.7 as
m H increases from zero to infinity, taking the value 2.1 when m H = 2.8m w. In
the range 25 Ge V < m H < 250 Ge V, E is well approximated by
E(x) = 1.58 + 0.32x - O.05x 2 •
(4.187)
Allowing Ow ::f: 0 changes Emin by about 100 GeV.
Although the configuration has the minimum energy among those satisfying
the ansatz, it is nevertheless unstable against perturbations which do not satisfy
the ansatz. This was to be expected too from the schematic diagram in figure 4.9,
in which the top of the potential barrier is evidently a maximum. Because of this
instability, the configuration was named a 'sphaleron', from the Greek meaning
'ready to fall'. The energy of the sphaleron satisfies
8 TeV < Espb < 14 TeV
(4.188)
and measures the height of the saddle point in configuration space over which
the vacuum must be 'pushed' to reach a topologically distinct vacuum. This
suggests that the exponentially small tunnelling rate (4.174) might be evaded by
supplying 0( I 0 Te V) of energy, for example in a pn collision. Then some baryon