172 The very early universe
that symmetry is restored in the very early universe. Hence, if the mass of the Higgs
particle did not satisfy the inequality
m^2 H> 3 e^2 M^2 G/ 16 π^2 , (4.120)known as the Linde–Weinberg bound, the symmetry would remain unbroken and
the gauge bosons would be massless.
Let us consider a special case:m^2 R= 0 ,or equivalently,Veff′′(χ ̄= 0 )= 0 .Then
it follows from (4.118) that potential (4.117) reduces to
Veff=
3 e^4
32 π^2(
χ ̄^4 ln
χ ̄
χ 0−
1
4
χ ̄^4 +1
4
χ 04)
, (4.121)
which is the Coleman–Weinberg potential. Such potentials may arise in unified
particle theories. They are especially interesting for cosmological applications and
can be used to construct the so called new inflationary scenario.
Now we derive the asymptotic behavior of the potential in the limit of very high
temperatures. To calculateF−for large temperatures,TmG(χ ̄),one can use
the high-temperature expansion (3.44) forJ−(^1 ).Then, taking into account (4.118),
potential (4.116) reduces to
Veff(χ, ̄ T)λT
4χ ̄^4 −e^3
4 πTχ ̄^3 +m^2 T
2χ ̄^2 +R, (4.122)where
λT=m^2 H
2 χ 02+
3 e^4
16 π^2lnbT^2
(eχ 0 )^2, lnb=2ln4π− 2 C 3. 5 (4.123)is the effective coupling constant and
m^2 T =e^2
4(
T^2 −T 02
)
, T 02 =
2 m^2 H
e^2−
3 e^2 χ 02
4 π^2(4.124)
is the temperature-dependent mass. Formula (4.122) is applicable only ifmG=
eχ ̄Tor, in other words, for ̄χT/e.
Note that our investigation so far was based on the one-loop approximation.
Higher order corrections can modify a detailed structure of the effective potential
at small ̄χ. In particular, it can be shown that when account is taken of these
corrections, the cubic term in (4.122) should be multiplied by the coefficient 2/3.
This effect, however, goes beyond the scope of our consideration and will be ignored
in what follows.
Problem 4.20Using (3.47) forJ(−^1 )(α, 0 ),find the first few terms in the low-
temperature expansion of effective potential (4.116). (HintIn this case it is more
convenient to usem/Tand∞as the limits of integration in (4.111).Why?)