188 The very early universe
with the instanton, is the unstablestaticsolution which depends only onthe spatial
coordinatesx.
For transitions dominated by sphalerons, the decay rate per unit time and unit
volume is
Bexp(
−
Vsph
T)
, (4.162)
where
Vsph= 4 π∫(
1
2
(
dφ
dr) 2
+V
)
r^2 dr (4.163)is the energy (ormass) of the sphaleron and B∼O
(
T^4 ,...
)
is a further pre-
exponential factor.
Using the same reasoning as for particle escape, we find that forTVsph/SI
tunneling is more important and to estimate the vacuum decay rate one has to use
(4.157); otherwise, forTVsph/SIthe decay rate is given by (4.162).
Thus, to calculate the vacuum decay rate we have to find the solution of either
(4.156) or (4.161), depending on the temperature. One must usually resort to nu-
merical calculation; however, for a wide class of scalar field potentials it is possible
to find the explicit expression forwithout specifying the shape of the potential
V.
Thin wall approximationThe first integral of (4.156) is
1
2(
dφ
dr ̃) 2
−V=
∫∞
̃r3
r ̃′(
dφ
dr ̃′) 2
dr ̃′, (4.164)where the boundary conditionφ→0asr ̃→∞has been used.Taking into account
the other boundary condition(dφ/dr ̃)r ̃= 0 = 0 ,we obtain the useful relation
−V(φ(r ̃= 0 ))=∫∞
03
r ̃′(
dφ
dr ̃′) 2
dr ̃′. (4.165)Now let us assume that the instanton, which is a bubble in four-dimensional Eu-
clidean “spacetime,” has a thin wall. This means that inside the bubble of radiusRI
the scalar field is nearly constant and equal toφ(r ̃= 0 ).The fieldφchanges very
fast within the wall – a thin layer of width 2lRI– and tends to zero outside
the bubble (Figure 4.14). Hence the integrand in (4.165) differs significantly from
zero only inside the wall, and this gives the dominant contribution to the integral.
Returning to (4.164) we see that the integral on the right hand side is suppressed