**4.8. Baryogenesis**

The symmetry between particles and antiparticles [126, 127], firmly established in collider physics, naturally leads to the question of why the observed universe is composed almost entirely of matter with little or no primordial antimatter.

Outside of particle accelerators, antimatter can be seen in cosmic rays
in the form of a few antiprotons, present at a
level of around 10^{-4} in comparison with the number of protons
(for example see
[128]).
However, this proportion is
consistent with secondary antiproton production through accelerator-like
processes,
*p* + *p*
3*p* + ,
as the cosmic rays stream
towards us. Thus there is no evidence for primordial antimatter in
our galaxy. Also, if matter and antimatter galaxies were to coexist
in clusters of galaxies, then we would expect there to be a detectable
background of
-radiation
from nucleon-antinucleon annihilations within
the clusters. This background is not observed and so we conclude that
there is negligible antimatter on the scale of clusters (For a review of
the evidence for a baryon asymmetry see
[129].)

More generally, if large domains of matter and antimatter exist, then annihilations would take place at the interfaces between them. If the typical size of such a domain was small enough, then the energy released by these annihilations would result in a diffuse -ray background and a distortion of the cosmic microwave radiation, neither of which is observed.

While the above considerations put an experimental upper bound on the
amount of antimatter in the universe, strict quantitative estimates of
the relative abundances of baryonic matter and antimatter may also be
obtained from the standard cosmology. The baryon number density does
not remain constant during the evolution of the universe, instead
scaling like *a*^{-3}, where *a* is the cosmological
scale factor
[130].
It is therefore convenient to define the baryon asymmetry
of the universe in terms of the quantity

(164) |

defined earlier. Recall that the range of
consistent
with the deuterium and ^{3}He primordial abundances is

(165) |

Thus the natural question arises; as the universe cooled from early times to today, what processes, both particle physics and cosmological, were responsible for the generation of this very specific baryon asymmetry? (For reviews of mechanisms to generate the baryon asymmetry, see

As pointed out by Sakharov [131], a small baryon asymmetry may have been produced in the early universe if three necessary conditions are satisfied

- baryon number (
*B*) violation, - violation of
*C*(charge conjugation symmetry) and*CP*(the composition of parity and*C*), - departure from thermal equilibrium.

The first
condition should be clear since, starting from a baryon symmetric universe
with = 0,
baryon number violation must take place in order to evolve
into a universe in which
does not
vanish. The second Sakharov criterion is required
because, if *C* and *CP* are exact symmetries, one can prove
that the total rate for any process which produces an excess of baryons is
equal to the rate of the complementary process which produces an
excess of antibaryons and so no net baryon number can be created.
That is to say that the thermal average of the baryon number operator
*B*, which is odd under
both *C* and *CP*, is zero unless those discrete symmetries are
violated. *CP* violation is present either if there are complex
phases in
the Lagrangian which cannot be reabsorbed by field redefinitions (explicit
breaking) or if some Higgs scalar field acquires a VEV which is not real
(spontaneous breaking). We will discuss this in detail shortly.

Finally, to explain the third criterion, one can calculate the equilibrium
average of *B* at a temperature *T* = 1 /
:

(166) |

where we have used that the Hamiltonian *H* commutes with
*CPT*. Thus
<*B*>_{T} = 0 in equilibrium and there is no
generation of net baryon number.

Of the three Sakharov conditions, baryon number violation and *C* and
*CP* violation may be investigated only within a given particle
physics
model, while the third condition - the departure from thermal equilibrium -
may be discussed in a more general way, as we shall see (for baryogenesis reviews see
[132,
133,
9,
10,
134,
135,
136].)
Let us discuss the Sakharov criteria in more detail.