CP-violation in electroweak theory
127
if we assume l ,.." t. However, the scale (4.204) is also the scale at which
perturbation theory breaks down in a hot plasma and it has been argued [47] that
plasma damping effects increase the time scale, so that
l
t - -
(4.206)
a2
basically because there are fewer ways large field configurations can cross the
barrier than smaller ones. This means that
rb
--a2 (U2T)4 -
(4.207)
V
2rr
and indeed a lattice simulation [48] gives
rb = (29±6)a~T4.
(4.208)
V
Numerically there is no difference between the two expressions.
At any rate, it is clear that there is no Boltzmann suppression at high
temperatures, so we do have a source of baryon-number non-conservation.
Whether or not it can explain the observed baryon asymmetry of the universe,
of course, depends upon the other two Sakharov criteria: the amount of CPviolation and whether the system is out of thennal equlibrium. The picture we
have in mind is that, in some region of space, there is a non-trivial gauge and
Higgs field configuration, of the type we have discussed, which leads to the nonconservation of baryon number. To generate a baryon asymmetry we require CPviolating interactions involving the quark fields. Then the transition rate for the
sphaleron-induced process with llNcs = + I will differ (slightly) from that with
llNc s = -I. Provided that the system is not in thennal equilibrium, there is then
the possibility of a net non-zero baryon-number asymmetry.
4.9 CP-violation in electroweak theory
In electroweak theory the sole source of CP-violation is via the CabibboKobayashi-Maskawa (CKM) matrix, which derives a CP-violating phase from the
unremovable phases in the Yukawa interactions of the quarks when we transfonn
to the mass eigenstates. The fonn of these interactions is
£y = q/iiRhDQL + Y,tURhUQL +h.c.
(4.209)
where QL is the quark doublet and UR, dR the singlets, tP is the Higgs scalar
doublet and y, = i1"2tP*. As in equation (4.70), hu and ho are complex matrices
acting on the undisplayed generation indices of QL. UR. dR. Also. as in equation
(4.78), we need to construct a diagram with non-vanishing imaginary parts in both
the loop integral and the trace over generation indices. It is easy to see that (4.81)
127
if we assume l ,.." t. However, the scale (4.204) is also the scale at which
perturbation theory breaks down in a hot plasma and it has been argued [47] that
plasma damping effects increase the time scale, so that
l
t - -
(4.206)
a2
basically because there are fewer ways large field configurations can cross the
barrier than smaller ones. This means that
rb
--a2 (U2T)4 -
(4.207)
V
2rr
and indeed a lattice simulation [48] gives
rb = (29±6)a~T4.
(4.208)
V
Numerically there is no difference between the two expressions.
At any rate, it is clear that there is no Boltzmann suppression at high
temperatures, so we do have a source of baryon-number non-conservation.
Whether or not it can explain the observed baryon asymmetry of the universe,
of course, depends upon the other two Sakharov criteria: the amount of CPviolation and whether the system is out of thennal equlibrium. The picture we
have in mind is that, in some region of space, there is a non-trivial gauge and
Higgs field configuration, of the type we have discussed, which leads to the nonconservation of baryon number. To generate a baryon asymmetry we require CPviolating interactions involving the quark fields. Then the transition rate for the
sphaleron-induced process with llNcs = + I will differ (slightly) from that with
llNc s = -I. Provided that the system is not in thennal equilibrium, there is then
the possibility of a net non-zero baryon-number asymmetry.
4.9 CP-violation in electroweak theory
In electroweak theory the sole source of CP-violation is via the CabibboKobayashi-Maskawa (CKM) matrix, which derives a CP-violating phase from the
unremovable phases in the Yukawa interactions of the quarks when we transfonn
to the mass eigenstates. The fonn of these interactions is
£y = q/iiRhDQL + Y,tURhUQL +h.c.
(4.209)
where QL is the quark doublet and UR, dR the singlets, tP is the Higgs scalar
doublet and y, = i1"2tP*. As in equation (4.70), hu and ho are complex matrices
acting on the undisplayed generation indices of QL. UR. dR. Also. as in equation
(4.78), we need to construct a diagram with non-vanishing imaginary parts in both
the loop integral and the trace over generation indices. It is easy to see that (4.81)