198 The very early universe
andfgoes from 4 to 12 for three quark families, each of which is represented in
three different colors. From (4.196) and (4.183 ), it follows that during a topological
transition in which the winding number increases byνunits, the fermion number
ineachdoublet decreases byνunits and hence the total fermion number is not
conserved. Taking into account that there are nine quark doublets and that the
baryon number of every quark is equal to 1/ 3 ,we obtain the following selection
rule:
Le=Lμ=Lτ=^13 B, (4.198)whereLiis the change of the lepton number in the corresponding family andB
is an overall change of the baryon number. Of course, the conservation laws for
energy, total electric charge and color should be fulfilled. In an instanton/sphaleron
transition the total number of fermions decreases by twelve units; correspondingly
the total lepton and baryon numbers decrease by three units each:L=B=− 3.
The energy of the disappearing fermions is transferred to the remaining and newly
created fermions and antifermions. One of the possible processes of this kind is
shown in Figure 4.18.
Thus in chiral theories topological transitions lead to nonconservation of the
total number of left-handed fermions. In electroweak theory there exist interactions
which convert left-handed particles to right-handed ones. Hence the total fermion
number is not conserved and some linear combination of baryon and lepton number,
B+aL,should vanish at thermal equilibrium. The numerical coefficientais of
order unity and its exact value can be found taking into account the conservation
laws and analyzing the conditions for the chemical equilibrium of all particles
involved. In the Standard Model with three generations of fermions and one Higgs
d c c s t t b ν ν r
udI+
+
Fig. 4.18.