Physical Foundations of Cosmology

196 The very early universe

This estimate is in good agreement with more elaborate calculations according to
whichMsph7–13 TeV. As in the case of metastable vacuum decay, the sphaleron
sizeRsphis comparable to the instanton sizeρm∼M−W^1.
At high temperatures the rate of topological transitions is proportional to
exp(−Msph/T) and one expects that atT>10 TeV they are no longer suppressed.
In fact, the transitions become very efficient at much smaller temperatures. We
have found that the expectation value of the Higgs field, and hence the masses of
the gauge bosons, decrease as the temperature increases. As a consequence the
height of the barrier, proportional toMW(T),also decreases. At the moment when
MW(T)∼αwT,the exponential suppression,

exp

(

−

Msph(T)

T

)

∼exp

(

− 2 π

MW(T)

αwT

)

,

disappears and the rate of transitions per unit volume per unit time is

∼R−sph^4 ∼(αwT)^4. (4.191)

This estimate, based on dimensional grounds, is also roughly valid at higher temper-
atures where symmetry is restored, the gauge bosons are massless and the barrier
disappears. In the absence of the barrier there are no sphalerons and transitions
occur via field configurations of typical size∼(αwT)−^1. In electroweak theory the
symmetry is restored when the temperature exceeds∼100 GeV (see (4.142)) and
the topological transitions are very efficient. This leads to nonconservation of the
total fermion number.

4.5.4 Chiral anomaly and nonconservation of the fermion number

Chiral anomalyThe gauge interactions of the massless fermions preserve both
left- and right-handed currents,JLμ≡ψ ̄LγμψLandJRμ≡ψ ̄RγμψR, at the classical
level. This means that the fermion numbers (equal to the difference between the
numbers of fermions and antifermions) are conserved for each helicity separately.
As a consequence, both the total currentJμ=JLμ+JRμand the chiral currentJ 5 μ=
JRμ−JLμare conserved. In electrodynamics the conservation of the total current is
equivalent to charge conservation and its violation would be a disaster. On the other
hand, the violation of the chiral current would have no dramatic consequences.
In fact, for massive fermions the chiral current is not conserved even classically.
Quantum fluctuations lead to a violation of chiral current conservation for massless
fermions also. In quantum electrodynamics the triangle diagram, shown in Figure
4.17, induces thechiral anomaly:

∂μ

(

JRμ−JLμ

)

=

e^2

8 π^2

FF ̃. (4.192)