48
Phase transitions in the earJy universe
where, again, we have retained only the top-quark Yukawa coupling. Provided
e 2 » A » e 6 , we estimate
r. 2 '" _2m 2
(2.104)
c -
•
As in the Higgs model, the shifted masses for the Higgs scalars are of order
_m 2 and the high-temperature approximation is not too unreasonable. The
greatest danger for the high-temperature approximation in the Higgs model with
e 4 » A arose from the gauge field mass. In the present case, dropping A/2
in the numerator of (2.103) and using the value of tP; at the zero-temperature
asymmetric minimum tP; = _m 2 lA, we can estimate that
r. 2
A
r. 2
A
_c_ '" 2--I- ~ 1.5'2'
(2.105)
m2 - e2
e
w
m z
Thus, if A ~ e 2 ~ 0.09, the high-temperature approximation may again be not
too unreasonable.
This discussion suggests that the electroweak phase transition is effectively
second order, because C, defined in (2.45), is small in the sense discussed at the
end of section 2.4.1. For T > Tc. the system is in the symmetric phase in which
tPc = 0 and all gauge bosons are massless. For T < Te, the system is in the
asymmetric phase for which tPc ". 0, the W± and Z gauge bosons acquire a
mass and the symmetry is broken from SU(2)L x U(l)y to U(l)em. The critical
temperature Tc given by (2.103) is of the same order of magnitude as the zerotemperature value of tPc at the asymmetric minimum of the effective potential
provided that A ~ e 2 . It was estimated before (2.102) that tPc ~ 263 GeV and so
Te should be of this order of magnitude.
2.6 Phase transitions in grand unified theories
Electroweak theory combines the weak and electromagnetic interactions in a
single model with SU(2)L x U(l)y gauge group but achieves no unification
of these interactions with the strong interaction. It is possible that the weak,
electromagnetic and strong interactions are unified in a theory involving a larger
gauge group (a grand unified theory or GUT), perhaps with a single gauge
coupling constant. Once such a unification has been assumed, the coupling
constants g, g' and gs (the QCD coupling constant) are related by group theory
factors to a single GUT coupling constant gG for the grand unified group. The
values of the renormalized coupling constants depend on the renormalization
scale M and, if the coupling constants geM), g' (M) and gs(M) obey the gauge
theoretic relationships of the grand unified group at one such scale M = M G, they
cannot obey these relationships at lower energy scales. This is because at energy
scales below MG the extra gauge fields associated with the enlargement of the
gauge group to the grand unified group may be ignored. (They acquire masses on
the scale of MG.) Then geM), g'(M) and gs(M) run differently with M when the
Phase transitions in the earJy universe
where, again, we have retained only the top-quark Yukawa coupling. Provided
e 2 » A » e 6 , we estimate
r. 2 '" _2m 2
(2.104)
c -
•
As in the Higgs model, the shifted masses for the Higgs scalars are of order
_m 2 and the high-temperature approximation is not too unreasonable. The
greatest danger for the high-temperature approximation in the Higgs model with
e 4 » A arose from the gauge field mass. In the present case, dropping A/2
in the numerator of (2.103) and using the value of tP; at the zero-temperature
asymmetric minimum tP; = _m 2 lA, we can estimate that
r. 2
A
r. 2
A
_c_ '" 2--I- ~ 1.5'2'
(2.105)
m2 - e2
e
w
m z
Thus, if A ~ e 2 ~ 0.09, the high-temperature approximation may again be not
too unreasonable.
This discussion suggests that the electroweak phase transition is effectively
second order, because C, defined in (2.45), is small in the sense discussed at the
end of section 2.4.1. For T > Tc. the system is in the symmetric phase in which
tPc = 0 and all gauge bosons are massless. For T < Te, the system is in the
asymmetric phase for which tPc ". 0, the W± and Z gauge bosons acquire a
mass and the symmetry is broken from SU(2)L x U(l)y to U(l)em. The critical
temperature Tc given by (2.103) is of the same order of magnitude as the zerotemperature value of tPc at the asymmetric minimum of the effective potential
provided that A ~ e 2 . It was estimated before (2.102) that tPc ~ 263 GeV and so
Te should be of this order of magnitude.
2.6 Phase transitions in grand unified theories
Electroweak theory combines the weak and electromagnetic interactions in a
single model with SU(2)L x U(l)y gauge group but achieves no unification
of these interactions with the strong interaction. It is possible that the weak,
electromagnetic and strong interactions are unified in a theory involving a larger
gauge group (a grand unified theory or GUT), perhaps with a single gauge
coupling constant. Once such a unification has been assumed, the coupling
constants g, g' and gs (the QCD coupling constant) are related by group theory
factors to a single GUT coupling constant gG for the grand unified group. The
values of the renormalized coupling constants depend on the renormalization
scale M and, if the coupling constants geM), g' (M) and gs(M) obey the gauge
theoretic relationships of the grand unified group at one such scale M = M G, they
cannot obey these relationships at lower energy scales. This is because at energy
scales below MG the extra gauge fields associated with the enlargement of the
gauge group to the grand unified group may be ignored. (They acquire masses on
the scale of MG.) Then geM), g'(M) and gs(M) run differently with M when the