180 The very early universe
and the symmetry is broken. The transition is a cross-over with no dramatic cos-
mological consequences; in particular, no large deviation from the thermal equilib-
rium. The temperature at transition depends on the Higgs mass and it follows from
(4.137) that, for instance,T 0 166 GeV formH120 GeV andT 0 240 GeV
formH180 GeV.The aboveestimatesare in good agreement with the results
of more rigorous and elaborate calculations.
4.5 Instantons, sphalerons and the early universe
In gauge theories which incorporate the Higgs mechanism the vacuum has a non-
trivial structure in several respects. First of all, as we have seen in the preceding
section, in the early universe the effective potential of the Higgs field can have two
local minima. If the transition from the symmetrical phase to the broken symmetry
phase happens when the false and true vacua are still separated by a barrier, the
transition is first order and accompanied by bubble nucleation. Although this sit-
uation seems not to occur in either quantum chromodynamics or the electroweak
model, it is rather typical for unified field theories beyond the Standard Model.
Another interesting aspect of non-Abelian gauge theories is the existence of
topologically different vacua. They are also separated by a barrier and topological
transitions can take place in the early universe. These transitions are very important
because they lead to anomalous nonconservation of fermion number in the Standard
Model.
If the temperature at the time of transition is small compared to the height of the
potential barrier, the transitions between different vacua occur as a result of subbar-
rier quantum tunneling. In this case the Euclidean solution of the field equations,
calledan instanton, gives the dominant contribution to the tunneling probability.
On the other hand, if the temperature is high enough, thermal fluctuations can take
the field over the barrier to the other vacuum without tunneling. In this situation the
transitions are classical and their rate is determined by the static field configuration
corresponding to the maximum of the potential, calleda sphaleron.
In this section we will calculate the rates for the false vacuum decay and for topo-
logical vacuum transitions and determine under which conditions the sphalerons
dominate the instantons and vice versa.
4.5.1 Particle escape from a potential well
To start with, let us consider as a “warm-up” the one-dimensional problem of a
particle of massMescaping from a potential well atq 0 =0 (Figure 4.12). In this
simple case we meet the main concepts we need for analyzing transitions in field
theory.