4.4 “Symmetry restoration” and phase transitions 171where the integralIis defined in (4.104) andVGis the energy density of the vacuum
fluctuations of the vector field with mass
mG(χ ̄)=eχ. ̄The factor 3 in (4.115) is due to the fact that the massive vector field has three degrees
of freedom at every point in space. The calculation of the temperature-dependent
contribution of the vector field essentially repeats the calculation for the scalar field
and the final result,which includes both quantum and thermal contributions, is
Veff(χ, ̄ T)=V+3 m^4 G(χ ̄)
64 π^2lnm^2 G(χ ̄)
μ^2+
3 T^4
4 π^2F−
(
mG(χ ̄)
T)
, (4.116)
wheremG(χ ̄)=eχ ̄andF−is defined in (4.111).
At zero temperature the last term vanishes and for the quartic potential in ( 4.107)
we obtain the following result:
Veff=
λR
4χ ̄^4 +
m^2 R
2χ ̄^2 +R+
3 e^4
32 π^2χ ̄^4 ln
χ ̄
χ 0, (4.117)
where the renormalized constantsλR,mR,Rcan be expressed through experi-
mentally measurable parameters. The concrete set of these parameters depends on
the normalization conditions used.
Problem 4.19Assume that the potentialVeffhas its minimum at someχ 0 = 0
and is equal to zero at this minimum, that is, there is no cosmological constant in
the broken symmetry phase. Solving the equationsVeff(χ 0 )= 0 ,Veff′(χ 0 )=0 and
Veff′′(χ 0 )=m^2 H,verify that
λR =m^2 H
2 χ 02−
9 e^4
32 π^2
, m^2 R=−m^2 H
2+
3 e^4 χ 02
16 π^2,
R=
χ 02
4[
m^2 H
2−
3 e^4 χ 02
32 π^2]
.
(4.118)
Thus we have expressed the renormalized constants in (4.117) in terms ofχ 0 ,the
gauge coupling constanteand the Higgs massmH.
Given that in the broken symmetry phaseMG≡mG(χ 0 )=eχ 0 , (4.119)we note that ifm^2 H< 3 e^2 M^2 G/ 8 π^2 ,thenm^2 R> 0 ,and the potential (4.117) ac-
quires a second local minimum at ̄χ= 0 .Moreover, form^2 H< 3 e^2 M^2 G/ 16 π^2 ,this
minimum is even deeper than the minimum atχ 0 ,becauseVeff(χ ̄= 0 )=R< 0.
Therefore symmetry breaking becomes energetically unfavorable. We will see later