Physical Foundations of Cosmology

170 The very early universe

In deriving this formula, we have changed the integration variablek→ωk/Tto
express the result through the integralJ(−^1 )defined in (3.34). For thermal fluctuations
the third term on the left hand side in (4.94) can be rewritten as

1

2

V′′′

〈

φ^2

〉

T=

∂mφ

∂χ ̄

mφ

T^2

4 π^2

J−(^1 )=

∂VφT

∂χ ̄

, (4.110)

where

VφT=

T^4

4 π^2

m∫φ/T

0

αJ−(^1 )(α, 0 )dα≡

T^4

4 π^2

F−

(mφ

T

)

(4.111)

is the temperature-dependent contribution of scalar particles toVeff.
The final result,which includes both quantum and thermal contributions, is

Veff=V+

m^4 φ(χ ̄)

64 π^2

ln

m^2 φ(χ ̄)

μ^2

+

T^4

4 π^2

F−

(

mφ(χ ̄)

T

)

, (4.112)

wherem^2 φ(χ ̄)=V′′(χ ̄).

4.4.2 U( 1 )model

Now we calculate the effective potential in aU( 1 )gauge model. The equation for
the scalar field immediately follows from (4.44),

χ;α;α+V′(χ)−e^2 χGμGμ= 0. (4.113)

In this case the calculation of the contribution of the scalar particles toVeffis a bit
more complicated because the fieldχis unambiguously defined only forχ> 0.
To avoid the complications we consider, therefore, only the most interesting case
when the contribution of the vector particles dominates that of the scalar particles.
For the quartic potential in (4.107) this meanse^2 λ,that is, the mass of the
gauge bosonmG(χ ̄)=eχ ̄is much bigger than the mass of the Higgs particle. Note
that the calculation of the one-loop contribution of the vector particles toVeffcan
still be trusted, even when it becomes comparable to theλχ ̄^4 term. Neglecting the
contribution of fieldφ,we find that the homogeneous component of the scalar field
satisfies the equation

χ ̄;α;α+V′(χ ̄)−e^2 χ ̄

〈

GμGμ

〉

= 0. (4.114)

The term

〈

GμGμ

〉reg

vaccan be calculated similarly to

〈

φ^2

〉reg
vacand it is easy to show
that

−e^2 χ ̄

〈

GμGμ

〉reg

vac=

∂

∂χ ̄

(

3 I(mG(χ ̄))

4 π^2

)

≡

∂VG

∂χ ̄

, (4.115)