Phase transitions and electroweak baryogenesis
129
With A .... UP perhaps to allow for the large numberof Feynman diagrams, we
still have
8sm :s 10- 25
(4.219)
using the current data [7]. This tiny number scales the difference in free energies
and, hence, the difference between the rates of the fl.Ncs = ±I processes.
By itself. it is sufficient to exclude the possibility of explaining the observed
baryon number asymmetry of the universe in the context of the standard model
electroweak theory. so extra sources of CP-violation are clearly needed. Such
sources arise naturally in the supersymmetric version of the theory which we
shall discuss shortly. Before doing so, however, we shall see that there are further
reasons why the standard model electroweak theory cannot yield the measured
asymmetry.
4.10 Phase transitions and electroweak baryogenesis
We have seen that electroweak theory possesses (sphaleron-induced) baryonnumber non-conserving processes. as well as CP-violation via the CKM matrix.
However, there is a rather general argument that these cannot generate the baryon
asymmetry of the universe in the absence of a phase transition or if the phase
transition is second order [51].
Suppose the latter and consider the universe at temperature T satisfying
mw < T < Tc
(4.220)
with Tc the critical temperature of the phase transition. For a second-order phase
transition
(
2 ) I /2
mw(T) = mw 1 - ~c2
(4.221)
We can see this from (4.189) by setting C to zero. Then v(T) and. hence, m w(T),
approach zero continuously as T approaches Tc from below; also, when C = 0,
To = TI = Tc. As in the case of GUT baryogenesis (4.37). we now require
that the rate r sph of baryon-number non-conserving processes is smaller than the
(Hubble) rate H(T) associated with the expansion of the universe, so that the
baryons are decoupled from the thermal bath. The total sphaleron rate r s~h is
obtained by scaling the rate per unit volume r sphl V, given in (4.201), with R (I),
where R(t) is the scale factor. proportional to T- 1 in the radiation-dominated era.
Thus, roughly,
ESPh(T)]
rsph .... kTexp [ -
T
(4.222)
where Esph(T) is given in (4.197). With H (T) given by (4.21), decoupling only
happens when the temperature T drops below T* with
2]-1/2
T*
Tc a2
mp
[
(4.223)
- ~ 1 + (--In-)
Tc
mw 2£
T*
129
With A .... UP perhaps to allow for the large numberof Feynman diagrams, we
still have
8sm :s 10- 25
(4.219)
using the current data [7]. This tiny number scales the difference in free energies
and, hence, the difference between the rates of the fl.Ncs = ±I processes.
By itself. it is sufficient to exclude the possibility of explaining the observed
baryon number asymmetry of the universe in the context of the standard model
electroweak theory. so extra sources of CP-violation are clearly needed. Such
sources arise naturally in the supersymmetric version of the theory which we
shall discuss shortly. Before doing so, however, we shall see that there are further
reasons why the standard model electroweak theory cannot yield the measured
asymmetry.
4.10 Phase transitions and electroweak baryogenesis
We have seen that electroweak theory possesses (sphaleron-induced) baryonnumber non-conserving processes. as well as CP-violation via the CKM matrix.
However, there is a rather general argument that these cannot generate the baryon
asymmetry of the universe in the absence of a phase transition or if the phase
transition is second order [51].
Suppose the latter and consider the universe at temperature T satisfying
mw < T < Tc
(4.220)
with Tc the critical temperature of the phase transition. For a second-order phase
transition
(
2 ) I /2
mw(T) = mw 1 - ~c2
(4.221)
We can see this from (4.189) by setting C to zero. Then v(T) and. hence, m w(T),
approach zero continuously as T approaches Tc from below; also, when C = 0,
To = TI = Tc. As in the case of GUT baryogenesis (4.37). we now require
that the rate r sph of baryon-number non-conserving processes is smaller than the
(Hubble) rate H(T) associated with the expansion of the universe, so that the
baryons are decoupled from the thermal bath. The total sphaleron rate r s~h is
obtained by scaling the rate per unit volume r sphl V, given in (4.201), with R (I),
where R(t) is the scale factor. proportional to T- 1 in the radiation-dominated era.
Thus, roughly,
ESPh(T)]
rsph .... kTexp [ -
T
(4.222)
where Esph(T) is given in (4.197). With H (T) given by (4.21), decoupling only
happens when the temperature T drops below T* with
2]-1/2
T*
Tc a2
mp
[
(4.223)
- ~ 1 + (--In-)
Tc
mw 2£
T*