38
Phase transitions in the early universe
The zero-temperature one-loop correction
of (2.27) and (2.23) has
contributions proportional to A 2 and e 4 from different loop diagrams. Provided
that A is small and that, in addition, e 4 « A, v: may be neglected compared to
the tree tenn (2.42). In the high-temperature limit,
T2 » At/>: e
2.1. 2
_m 2
'l'c
(2.43)
the one-loop temperature-dependent contribution to the effective potential Vr
obtained from (2.32) is given by
-T
4",2T 4
(A. +
= -90 +
3e 2 )T2 2 CT 3
VI (~c)
24
~c - 3~c + ...
(2.44)
where
41fC = tr[Mi(~c»3/2 + 3 tr[M~(~c»3/2
3A )3 / 2 (
A )3 / 2
= ( m2~;2 + 4" + m2~;2 + 4' + 3e 3
(2.45)
~ ( - 3A)3/2 + (A)3/2 - +3e 3
(2.46)
4
4
when A~,! » m 2 • Thus, the complete effective potential to one-loop order is given
by
-
= -90
4",2T 4 +"2
I 2
2
CT 3
A 4
V(~c)
m (T)t;c - 3~c + 16~c
(2.47)
where a temperature-dependent mass m 2 (T) has been defined by
2(T)
2
(A + 3e 2 )T2
m
=m+-'---'-(2.48)
12
The expectation value ~c of the scalar field is obtained by minimizing the
effective potential. For sufficiently high temperatures, there is only one solution
of
av =0
(2.49)
a~c
namely
~c =0
(2.50)
and this is a minimum so long as m 2 (T) is positive. From (2.48), we see that this
is the case provided the temperature T exceeds To where
T,2 = -12m2
(2.5t)
o - A+3e2 ·
(See figure 2.1, curve A.) We may write
v: