**4.9. Baryon Number Violation**

**4.9.1. B-violation in Grand Unified
Theories**

As discussed earlier, Grand Unified Theories (GUTs)
[137]
describe the fundamental
interactions by means of a unique gauge group *G* which contains the
Standard Model (SM) gauge group *SU*(3)_{C}
*SU*(2)_{L}
*U*(1)_{Y}.
The fundamental idea of GUTs is that at energies higher than a certain
energy threshold
*M*_{GUT} the group symmetry is *G* and that, at
lower energies, the symmetry is broken down to the SM gauge symmetry,
possibly through a chain of symmetry breakings.
The main motivation for this scenario is that, at least in supersymmetric
models, the (running) gauge couplings of the SM unify
[138,
139,
140]
at the scale
*M*_{GUT}
2 × 10^{16} GeV, hinting at the presence
of a GUT involving a higher symmetry with a single gauge coupling.

Baryon number violation seems very natural in GUTs. Indeed,
a general property of these theories is that the same representation of
*G* may contain both quarks and leptons, and therefore it is
possible for scalar and gauge bosons
to mediate gauge interactions among fermions having different baryon
number.

**4.9.2. B-violation in the Electroweak
theory.**

It is well-known that the most general renormalizable Lagrangian invariant
under the SM gauge group and containing only color singlet Higgs fields
is automatically invariant under global abelian
symmetries which may be identified with the baryonic and leptonic
symmetries.
These, therefore, are accidental symmetries and as a result it is not
possible to violate *B* and *L* at tree-level or at any order of
perturbation theory. Nevertheless, in many cases the perturbative expansion
does not describe all the dynamics of the theory and, indeed, in 1976 't
Hooft [141]
realized that nonperturbative effects (instantons) may give rise to
processes which violate the combination *B* + *L*, but not the
orthogonal combination *B* - *L*. The probability of these
processes occurring today is exponentially
suppressed and probably irrelevant. However, in more extreme situations -
like the primordial universe at very high temperatures
[142,
143,
144,
145]
- baryon and lepton number violating processes may be fast enough to
play a significant role in baryogenesis. Let us have a closer look.

At the quantum level, the baryon and the lepton symmetries are
anomalous [146,
147],
so that their respective Noether currents
*j*_{B}^{µ} and
*j*_{L}^{µ} are no longer conserved, but satisfy

(167) |

where *g* and *g'* are the gauge couplings of
*SU*(2)_{L} and *U*(1)_{Y},
respectively, *n*_{f} is the number of families and
^{µ} =
(1/2)^{µ}
*W*_{
}
is the dual of the *SU*(2)_{L} field strength tensor, with
an analogous expression holding for
. To understand how
the anomaly is closely
related to the vacuum structure of the theory, we may compute the change in
baryon number from time *t* = 0 to some arbitrary final time
*t* = *t*_{f}.
For transitions between vacua, the average values of the field strengths
are zero at the beginning and the end of the evolution. The change in
baryon number may be written as

(168) |

where the Chern-Simons number is defined to be

(169) |

where *A*_{i} is the SU(2)_{L} gauge field.
Although the Chern-Simons number is not gauge invariant, the change
*N*_{CS} is. Thus, changes in Chern-Simons number result in
changes in baryon number which are integral multiples of the number of
families *n*_{f} (with *n*_{f} = 3 in the real
world).
Gauge transformations *U*(*x*) which connects two degenerate
vacua of the gauge theory may change the Chern-Simons number by an integer
*n*, the winding number.

If the system is able to perform a transition from the vacuum
_{vac}^{(n)}
to the closest one _{vac}^{(n±1)}, the Chern-Simons number is
changed by unity and
*B* =
*L* =
*n*_{f}.
Each transition creates 9 left-handed quarks (3 color states for each
generation) and 3 left-handed leptons (one per generation).

However, adjacent vacua of the electroweak theory are separated by a
ridge of
configurations with energies larger than that of the vacuum.
The lowest energy point on this ridge is a saddle point solution
to the equations of motion with a single negative eigenvalue, and
is referred to as the *sphaleron*
[143,
144].
The probability of baryon number nonconserving processes at zero temperature
has been computed by 't Hooft
[141]
and is highly suppressed.

The thermal rate
of baryon number violation in the *broken* phase is

(170) |

where *µ* is a dimensionless constant.
Although the Boltzmann suppression in (170) appears large, it is to
be expected that, when the electroweak symmetry becomes restored at a
temperature of around 100 GeV, there will no longer be an exponential
suppression factor. A simple
estimate is that the rate per unit volume of sphaleron events is

(171) |

with another dimensionless constant. The rate of sphaleron processes can be related to the diffusion constant for Chern-Simons number by a fluctuation-dissipation theorem [148] (for a good description of this see [134]).

*CP* violation in GUTs arises in loop-diagram corrections to
baryon number violating bosonic decays. Since it is necessary that
the particles in the loop also undergo
*B*-violating decays, the relevant particles are the *X*,
*Y*, and *H*_{3} bosons in the case of *SU*(5).

In the electroweak theory things are somewhat different.
Since only the left-handed
fermions are *SU*(2)_{L} gauge coupled, *C* is
maximally broken in the SM.
Moreover, *CP* is known not to be an exact symmetry
of the weak interactions. This is seen experimentally in the neutral
kaon system through *K*_{0},
_{0} mixing. Thus,
*CP* violation is a natural feature of the
standard electroweak model.

While this is encouraging for baryogenesis, it turns out that this
particular source of *CP* violation is not strong enough. The relevant
effects are parameterized by a dimensionless constant which is no
larger than 10^{-20}. This appears to be much too small to account
for the observed BAU and, thus far, attempts to utilize this source of
CP violation for electroweak baryogenesis have been unsuccessful. In
light of this, it is usual to extend the SM in some fashion
that increases the amount of *CP* violation in the theory while not
leading to results that conflict with current experimental data. One
concrete example of a well-motivated extension in the minimal
supersymmetric standard model (MSSM).

**4.9.4. Departure from Thermal Equilibrium**

In some scenarios, such as GUT baryogenesis, the third Sakharov condition is satisfied due to the presence of superheavy decaying particles in a rapidly expanding universe. These generically fall under the name of out-of-equilibrium decay mechanisms.

The underlying idea is fairly simple. If the decay rate
_{X} of the
superheavy particles *X* at the time they become
nonrelativistic (*i.e.* at the temperature *T* ~
*M*_{X}) is much smaller
than the expansion rate of the universe, then the *X*
particles cannot decay on the time scale of the expansion and so they
remain as abundant as photons for
*T*
*M*_{X}. In other words, at some temperature
*T* > *M*_{X}, the superheavy particles
*X* are so weakly interacting that they cannot catch up with the
expansion of the universe and they decouple from the thermal bath while
they are still relativistic, so that *n*_{X} ~
*n*_{} ~ *T*^{3} at the time of decoupling.

Therefore, at temperature
*T*
*M*_{X}, they populate the universe
with an abundance which is much larger than the equilibrium one.
This overabundance is precisely
the departure from thermal equilibrium needed to produce a final
nonvanishing baryon asymmetry when the heavy states *X* undergo
*B* and *CP* violating decays.

The out-of-equilibrium condition requires very heavy states:
*M*_{X}
(10^{15}
- 10^{16}) GeV and *M*_{X}
(10^{10}
- 10^{16}) GeV, for gauge and scalar bosons, respectively
[149],
if these heavy particles decay through renormalizable operators.

A different implementation can be found in the electroweak theory. At temperatures around the electroweak scale, the expansion rate of the universe in thermal units is small compared to the rate of baryon number violating processes. This means that the equilibrium description of particle phenomena is extremely accurate at electroweak temperatures. Thus, baryogenesis cannot occur at such low scales without the aid of phase transitions and the question of the order of the electroweak phase transition becomes central.

If the EWPT is second order or a continuous crossover, the associated
departure from equilibrium is insufficient to lead to relevant baryon
number production
[145].
This means that for EWBG to succeed, we
either need the EWPT to be strongly first order or other methods of
destroying thermal equilibrium to be present at the phase transition.
For a first order transition there is an extremum at
= 0 which
becomes separated from a second local minimum by an
energy barrier. At the critical temperature *T* =
*T*_{c} both phases are equally favored energetically and
at later times the minimum at
0 becomes the global
minimum of the theory.

The dynamics of the phase transition in this situation is crucial to
most scenarios of electroweak baryogenesis. The essential picture is
that around *T*_{c} quantum tunneling occurs and nucleation
of bubbles
of the true vacuum in the sea of false begins. Initially these bubbles
are not large enough for their volume energy to overcome the competing
surface tension and they shrink and disappear. However, at a
particular temperature below *T*_{c}, bubbles just large
enough to grow
nucleate. These are termed *critical* bubbles, and they expand,
eventually filling all of space and completing the transition.

As the bubble walls pass each point in space, the order parameter changes rapidly, as do the other fields, and this leads to a significant departure from thermal equilibrium. Thus, if the phase transition is strongly enough first order it is possible to satisfy the third Sakharov criterion in this way.

A further natural way to depart from equilibrium is provided by the dynamics of topological defects [150, 151, 152, 153]. If, for example, cosmic strings are produced at the GUT phase transition, then the decays of loops of string act as an additional source of superheavy bosons which undergo baryon number violating decays.

When defects are produced at the TeV scale, a detailed analysis of the
dynamics of a network of these objects shows that a significant baryon
to entropy ratio can be generated if the electroweak symmetry is
restored around such a higher scale ordinary defect.
Although *B*-violation can be inefficient along nonsuperconducting
strings [154],
there remain
viable scenarios involving other ordinary defects, superconducting strings
or defects carrying baryon number.

**4.9.5. Baryogenesis via leptogenesis**

Since the linear combination *B* - *L* is left unchanged by
sphaleron transitions, the baryon asymmetry may be generated from a lepton
asymmetry
[155,
156,
157] (see also
[158,
159].)
Indeed, sphaleron transition will reprocess any
lepton asymmetry and convert (a fraction of) it into baryon
number. This is because *B* + *L* must be vanishing and the
final baryon asymmetry results to be
*B* - *L*.

In the SM as well as in its unified extension based on the group
*SU*(5), *B* - *L* is conserved and no asymmetry in
*B* - *L* can be
generated. However, adding right-handed Majorana neutrinos to the SM
breaks *B* - *L* and the primordial lepton asymmetry may be
generated by
the out-of-equilibrium decay of heavy right-handed Majorana neutrinos
*N*_{L}^{c} (in the supersymmetric version, heavy
scalar neutrino decays
are also relevant for leptogenesis). This simple extension of the SM
can be embedded into GUTs with gauge groups containing *SO*(10). Heavy
right-handed Majorana neutrinos can also explain the smallness of the
light neutrino masses via the see-saw mechanism
[157].

**4.9.6. Affleck-Dine Baryogenesis**

Finally in this section, we mention briefly a mechanism introduced by Affleck and Dine [160] involving the cosmological evolution of scalar fields carrying baryonic charge.

Consider a colorless, electrically neutral combination of quark and lepton fields. In a supersymmetric theory this object has a scalar superpartner, , composed of the corresponding squark and slepton fields.

An important feature of supersymmetric field theories is the existence of "flat directions" in field space, on which the scalar potential vanishes. Consider the case where some component of the field lies along a flat direction. By this we mean that there exist directions in the superpotential along which the relevant components of can be considered as a free massless field. At the level of renormalizable terms, flat directions are generic, but supersymmetry breaking and nonrenormalizable operators lift the flat directions and sets the scale for their potential.

During inflation it is likely that the
field is
displaced from the position
<> = 0,
establishing the initial conditions for
the subsequent evolution of the field. An important role is played at this
stage by baryon number violating operators in the potential
*V*(), which
determine the initial phase of the field. When the Hubble rate becomes
of the order of the curvature of the potential
~ *m*_{3/2}, the condensate starts
oscillating around its present minimum. At this time, *B*-violating
terms in the potential are of comparable importance to the mass term,
thereby imparting a substantial baryon number to the condensate. After this
time, the baryon number violating operators are negligible so that, when
the baryonic charge of
is transferred to fermions through decays, the net baryon number of the
universe is preserved by the subsequent cosmological evolution.

The most recent implementations of the Affleck-Dine scenario have been in the context of the minimal supersymmetric standard model [161, 162] in which, because there are large numbers of fields, flat directions occur because of accidental degeneracies in field space.