194 The very early universe
Thus, the field configuration interpolating between two topologically different
vacua has a nonvanishing field strength and hence, “in between,” nonzero posi-
tive potential energy. Because the energy of both the initial and final states is equal
to zero, the transition between the vacua can occur only as subbarrier tunneling
via an instanton. To find the corresponding instanton we make the Wick rotation
to Euclidean time:t→τ=itand substituteA 0 →iA 0 ; thenF 0 i→iF 0 iand the
form trF^2 becomes nonnegative definite. By the Schwartz inequality,
(∫
tr(
F^2
)
d^4 x)(∫
tr( ̃
F^2
)
d^4 x)
∣∣
∣∣
(∫
tr(
FF ̃
)
d^4 x)∣∣
∣∣
2. (4.184)
Taking into account that tr
(
F ̃^2
)
=tr(
F^2
)
, this inequality together with (4.183) im-
plies the lower bound for the Euclidean action of any field configuration connecting
the two vacua:
SE=1
2
∫
tr(
F^2
)
d^4 x8 π^2
g^2|ν 1 −ν 0 |. (4.185)The equality in (4.184) is attained if and only ifF=±F ̃.The corresponding inter-
polating solution withν=1 is called the instanton. We do not need the explicit
form of this solution here, but merely point out that it is characterized by a single
parameter (a constant of integration), the instanton sizeρ. The instanton action
does not depend on the size and for anyρis equal to
SI=8 π^2
g^2=
2 π
α(4.186)
whereα≡g^2 / 4 πis the corresponding “fine structure constant.”
Topological transitionsThus, the vacuum of a gauge theory generally has a com-
plicated structure with many minima separated by potential barriers as shown in
Figure 4.16. (This picture is, of course, no more than asymbolicrepresentation of
the vacuum structure and should not be taken too literally.) The instanton connects
two adjacent minima. Theprobabilityof tunneling is proportional to exp(− 2 SI)
because as opposed to the particle tunneling, the instanton interpolates between the
initial and final states only once. Hence the transition rate between topologically
different vacua is
∝exp(
−
4 π
α)
, (4.187)
In electroweak theory,αw 1 /29 and we have∝ 10 −^160. Therefore, instanton
transitions are strongly suppressed in electroweak theory.