212 The very early universe
XlqXlqFig. 4.21.violated, the rate(X→qq)≡ris generally not equal to
(
X ̄→q ̄q ̄)
≡r ̄.In
fact, let us assume that the only source forCPviolation is the Kobayashi–Maskawa
mechanism. Then the coupling constants can be complex: they enter the diagrams
describing the decay of particles, but their conjugated values enter the diagrams
for antiparticle decay. The difference in the decay rates can be seen in the inter-
ference of tree-level diagrams with higher-order diagrams such as in Figure 4.21.
This difference, characterizing the degree ofCPviolation, is proportional to the
imaginary part of the corresponding product of coupling constants and vanishes if
all constants are real-valued (CPis not violated).
One expects that at temperatures much higher thanmX, theXandX ̄bosons are in
equilibrium and their abundances are the same,nX=nX ̄∼nγ.As the temperature
drops to aboutmX,the processes responsible for maintaining equilibrium become
inefficient and the number density per comoving volume freezes out at some value
γ∗=nX/s(see Section 4.6.1). Subsequently, only theout of equilibrium decayof
theXandX ̄bosons is important and net baryon charge can be produced. Let us
estimate its size. We normalize theoveralldecay rate, which is the same for particles
and antiparticles, to unity. Then
(
X→q ̄ ̄l)
= 1 −r and(
X ̄→ql)
= 1 −r ̄.
The mean net baryon number produced in the decay of theXboson is
BX=( 2 / 3 )r+(− 1 / 3 )( 1 −r);likewise
BX ̄=(− 2 / 3 )r ̄+( 1 / 3 )( 1 −r ̄).Hence the resulting baryon asymmetry is
B=γ∗(BX+BX ̄)=γ∗(r−r ̄). (4.222)We see thatBdepends on the freeze-out concentrationγ∗and on a parameter
ε≡(r−r ̄)which characterizes the amount ofCPviolation. The termγ∗is mainly
determined by the rates of reactions responsible for equilibrium (see Section 4.6.1)
and does not exceed unity. The parameterεcomes from higher-order perturbation
theory and thereforeB 1 .For example, in the minimalSU( 5 )model the pa-
rameterεreceives its first nontrivial contribution at the tenth order of perturbation