4.4 “Symmetry restoration” and phase transitions 179retained in (4.135) only whenmt(χ ̄)=Mtχ/χ ̄ 0 T, that is, for ̄χ(χ 0 /Mt)T.
Within the range
T
MZ,Wχ 0 >χ> ̄T
Mtχ 0 , (4.138)theZandWbosons are relativistic, while thetquarks are nonrelativistic and hence
their contribution should be omitted. As we have mentioned above, for very small
χ ̄our derivation fails and we have to use more refined methods.
The analysis of the potential behavior follows almost exactly the consideration
in the previous section. At very high temperatures, namely
TT 1 =T 0
√
1 −
(
^2 / 4 ΥλTc), (4.139)
the potential has one minimum only, at ̄χ=0, the symmetry is restored and gauge
bosons and fermions are massless. As the temperature drops belowT 1 ,the second
minimum appears and at
Tc=T 0
√
1 −
(
2 ^2 / 9 ΥλTc) (4.140)
the depth of this second minimum, located at
χ ̄c=2 T
3 λTc, (4.141)
becomes the same as that of the minimum at ̄χ= 0 .Subsequently, the transition
to the broken symmetry phase becomes possible. As noted above, the minimum
at ̄χcis separated from ̄χ=0 by the barrier only if ̄χc/ 2 >T/
√
- Hence, we
expect a strong first order phase transition only ifλTc<
√
8 / 3 .Using (4.136),
this condition can be rewritten in terms of the upper bound on the Higgs mass:
mH<75 GeV.Thus, only if the Higgs particles were light enough would the
electroweak phase transition be first order. FormH=50 GeV,this would occur at
temperatureTc≈88 GeV.
The experimental bound on the Higgs mass ismH>114 GeV.Therefore in
reality we expect that the breaking of electroweak symmetry happens smoothly.
For large Higgs masses one can simply neglect the ̄χ^3 term in (4.135). In this case,
when the temperature drops belowT 0 the only minimum of the potential is located
at
χ ̄c=(
Υ
λT