126
Baryogenesis
with the upper bound saturated when m H achieves its lower bound. The
temperature-dependent sphaleron energy satisfies
Espb(T) = 2mw(T) £
(4.197)
a2
with
1.9 < t: < 2.7
(4.198)
and the associated Boltzmann factor exp[-Espb(T)/T] means that the baryonnumber non-conservation is unsuppressed when
T > Espb(T).
(4.199)
However, this is satisfied only when T is practically at the critical temperature at
which the energy barrier disappears in any case. Careful analytical estimates of
the baryon-number changing rates, valid in the range
2mw(T) « T « 2mw(T)
(4.200)
a2
have been made by Amold and McLerran [45]. They find a transition rate per unit
volume
rspb = K(2m W (T»4(2m W (T»)3 e-E .... (T)/T
(4.201)
V
a2T
where the numerical factor K '" 11, although a (non-perturbative) numerical
'measurement' of the diffusion rate of the Chem-Simons number over the barrier
[46] suggests that this may be too large by a factor of order 10.
At high temperatures where T > TI, the Higgs field VEV v(T) is zero
and there is no sphaleron. The gauge fields, however, can still generate baryonnumber non-conservation. The energy Eb of such a configuration is controlled by
the a priori length scale i of the configuration which changes the Chem-Simons
number Ncs. Presumably the scale of the action is set by (4.173), so
211'
Ebi'" - .
(4.202)
a2
To avoid Boltzmann suppression, we require
21r
Eb'" -
< T
(4.203)
a2i
so
i> 21r
(4.204)
"'a2 T'
Thus, the transition rate per unit volume is
rb '" _1_ '" (a 2T)4
(4.205)
V
i 3 ,
211'