Physical Foundations of Cosmology

4.4 “Symmetry restoration” and phase transitions 177

Problem 4.21Verify thatft〈tt ̄〉=∂Vt/∂χ, ̄ where

Vt≡ 3 × 4

(

−

I(mt)

4 π^2

+

T^4

4 π^2

F+

(mt

T

))

. (4.132)

Heremt=ftχ ̄and

F+

(mt

T

)

≡

m∫t/T

0

αJ(+^1 )(α, 0 )dα. (4.133)

The integralsI(mt)andJ(+^1 )(α, 0 )are defined in (4.104) and (3.34) respectively.
The factor 3 in (4.132) accounts for the three different colors oftquarks, the factor
4 for four degrees of freedom of the fermions of each color, and the negative sign
in front ofIindicates that the vacuum energy density of fermions is negative.

Using this result we obtain

Veff=V+

3

64 π^2

(

m^4 Zln

m^2 Z

μ^2

+ 2 m^4 Wln

m^2 W

μ^2

− 4 m^4 tln

m^2 t

μ^2

)

+

3 T^4

4 π^2

[

F−

(mZ

T

)

+ 2 F−

(mW

T

)

+ 4 F+

(mt

T

)]

, (4.134)

where

mZ=

√

g^2 +g′^2

2

χ, ̄ mW=

g

2

χ, ̄ mt=ftχ. ̄

This formula resembles (4.116) and the numerical coefficients in front of the differ-
ent terms can easily be understood by counting the number of degrees of freedom
of the corresponding fields.

Problem 4.22Using the same normalization conditions as in Problem 4.19, verify
that atT=0 K the effective potential is given by (4.117) and (4.118), where we
have to substitute

e^4 →

MZ^4 + 2 MW^4 − 4 M^4 t

χ 04

.

Since in the broken symmetry phase the masses of the gauge bosons andtquark are
MZ≡mZ(χ 0 ) 91 .2 GeV,MW 80 .4 GeV andMt170 GeV,this combina-
tion of masses is negative and the Linde–Weinberg arguments do not lead to a lower
bound on the Higgs mass. However, in this case the top quark contribution causes
the coefficient in front of the logarithmic term in (4.117) to be negative. At very
large ̄χ this term dominates and the potential becomes negative and unbounded
from below.