190 The very early universe
in Minkowski spacetime. The picture described is a very close analog of subbarrier
particle tunneling.
To understand when the thin wall approximation is valid we must determine
when the conditionl/RI1 is satisfied. IfVmis the height of thescalar field
potential, the positive energy inside the wall of the “emerging bubble” is of order
VmR^2 Il.This is exactly compensated by the negative energy inside the bubble
∼R^3 I,where−is the global minimum of the potential. Therefore,l/R∼
/Vmand hence the thin wall approximation is applicable only if/Vm1.
The thin wall sphaleron which determines the decay rate at very high tempera-
tures can be found in a similar way from (4.161).
Problem 4.25Verify that the thin wall sphaleron has sizeRsph≈ 2 σ/and mass
equal to
Vsph≈16 πσ^3
3 ^2. (4.171)
Thus, we find that if/Vm 1 ,the vacuum decay rate is aboutBexp(
−
16 πσ^3
3 ^2 T)
(4.172)
forTVsph/SI∼Rsph−^1 .On the other hand, forTRsph−^1 ∼R−I^1 ,the vacuum
decay rate is given by (4.169).
After a bubble has emerged in Minkowski spacetime its behavior can easily
be found if we analytically continue an appropriate solution of (4.156) back to
Minkowski spacetime. The corresponding functionφ(
√
r^2 −t^2 ) describes the ex-
panding bubble. The fieldφis constant along the hypersurfacesr^2 −t^2 =const.
Figure 4.15 shows these hypersurfaces and makes it clear that from the point of
view of an observer at rest, the thickness of the wall decreases with time and the
speed of the wall approaches the speed of light.
Problem 4.26Using the results of this section, derive and analyze the formulae
describing first order phase transitions in theU( 1 )model considered in Section
4.4.4.
4.5.3 The vacuum structure of gauge theories
In non-Abelian gauge theories the vacuum of gauge fields has a nontrivial structure.
To understand how this comes about, we first consider pureSU(N)theory without
fermions and scalar fields. The gauge fieldFμνshould vanish in the vacuum. This
does not mean, however, that the vector potentialAμalso vanishes; the vanishing of
Fμνonly means that the vector potential is a gauge transform of zero (see (4.11)).