Physical Foundations of Cosmology

4.4 “Symmetry restoration” and phase transitions 175

of the scalar field are about

δφ=

√〈

φ^2

〉

TT/

√

24.

For ̄χ<T/

√

24 ,the vector bosons can no longer be treated as massive particles.
Therefore, for this range,the perturbative consideration of the mass corrections in
J−(^1 )failsand one expects the ̄χ^3 term to be absent.
The following simple criteria provide a sense of when the barrier will be present:
if, at critical temperatureTc,the value of the scalar field at the expected location of
the barrier maximum,

χ ̄c/ 2 =e^3 Tc/ 4 πλT,

exceedsT/

√

24 ,then the barrier really exists.Indeed, in this case the calculation
of ̄χ^3 term is reliable. Thus, we conclude that if the coupling constantλis small
enough, namely
√
6
2 π

e^3 >λ>

9

16 π^2

e^4 , (4.129)

the maximum of the barrier is located at

Tc/e>χ ̄c/ 2 >Tc/

√

24

and the first order phase transition with bubble nucleation should take place. Because
the Higgs mass is proportional toλ,this situation can be realized only if the Higgs
particle is not too heavy.
On the other hand, if the coupling constant is large,

e^2 >λ>

√

6

2 π

e^3 , (4.130)

the barrier should be located at

χ ̄c/ 2 <Tc/

√

24.

However, for this range of ̄χ,the bosons should be treated as massless particles and
the ̄χ^3 term should be left out of the potential. Thus, we expect that the barrier does
not arise at all and the effective potential changes as shown in Figure 4.11. In this
case the symmetry breaking occurs smoothly via a gradual increase of the mean
value of the scalar field. Therefore, the transition has no dramatic cosmological
consequences. We remind the reader that the contribution of the scalar particles can
be neglected only if the first inequality in (4.130) is fulfilled.
The criteria derived above are no more than rough estimates. However, more
sophisticated analysis shows that these estimates reproduce the more rigorous re-
sults rather well.