Physical Foundations of Cosmology

4.4 “Symmetry restoration” and phase transitions 169

the mass is

m^2 φ=V′′= 3 λ 0 χ ̄^2 +m^20.

The divergent terms come to be multiplied with ̄χ^4 ,χ ̄^2 ,χ ̄^0 ,and can be combined
with appropriate terms inV(χ ̄).As a result, the bare constantsλ 0 ,m^20 and 0
are replaced by the finite renormalized constantsλR,m^2 R andR,measured in
experiments. The termV∞,therefore, can be omitted in (4.106).

Problem 4.17Find the explicit relations between the bare and renormalized pa-
rameters.

The potential (4.106) looks peculiar because it seems to depend explicitly on an
arbitrary scaleμ.However, it is easy to see that the change inμinduces a term
proportional tom^4 (χ ̄),which, for a renormalizable theory, has the same structure
as the original potentialVand hence leads to finite renormalization of constants.
These constants, therefore, become scale-dependent (running), reflecting the renor-
malization group properties of the quantum field theory. The physics remains the
same; it is only our way of interpreting the constants that changes. For pureχ^4
theory withm^2 φ(χ ̄)= 3 λχ ̄^2 ,we have

Veff=

1

4

λχ ̄^4 +

9 λ^2 χ ̄^4

32 π^2

ln

χ ̄

χ 0

, (4.108)

whereλ=λ(χ 0 )andχ 0 is some normalization scale. The logarithmic corrections to
the potential are proportional toλ^2 and become comparable to the leadingλχ ̄^4 term
only whenλln(χ/χ ̄ 0 )∼O( 1 ).At these large values of ̄χ,however, the higher-loop
contributions we have neglected thus far become crucial.

Problem 4.18Requiring that potential (4.108) should not depend onχ 0 ,derive
the renormalization group equation forλ(χ ̄). Solve this equation, keeping only the
leading term in theβfunction, and verify thatλ(χ ̄)blows up whenλ(χ 0 )ln(χ/χ ̄ 0 )∼
O( 1 ).

Thermal contribution to VeffIn a hot universe the fieldφis no longer in its vacuum
state. The occupation numbersnkare given by the Bose–Einstein formula (3.20),

where=ωk=

√

k^2 +m^2 φand the chemical potential can be neglected.Substitut-

ing the Bose–Einstein distribution in (4.102), we obtain the following expression
for the thermal contribution to

〈

φ^2

〉

:

〈

φ^2

〉

T=

1

2 π^2

∫∞

0

k^2 dk

ωk

(

eωk/T− 1

)=

T^2

4 π^2

J−(^1 )

(

mφ(χ ̄)

T

, 0

)

. (4.109)