50
Phase transitions in the early universe
where
24
A", == LAa",ta
(2.110)
a=l
with Aa", the gauge fields in the adjoint representation of S U (5), and the transition
to a finite-temperature theory is made by the replacement of a", by a"" as in (2.13).
The gauge field strength F",,, is given by
F",,, == a",A" - a"A", + igG[A"" A,,]
(2.111)
with the transition to the finite temperature being made in the same way. The
fermionic terms have been dropped in (2.108) because quark and lepton masses
are negligible on the grand unified scale.
For the breaking of SU(5) to SU(3)c x SU(2)L x U(l)Y, we take the
expectation value of the field cl> to be of the form
t;c . (
(cl» = v'I5 dlag I. I. I. -2' 3 -2 3) .
(2.112)
«(cl» must be traceless because the matrices ta are.) This can be shown to be the
lowest energy state at zero temperature for
7
AI> --A2
A2 > O.
(2.113)
30
The finite-temperature effective potential can then be written. for temperatures
large compared to all masses, as
- = 2 1 2 2 + 4 I ( 7) 4 ( 7) 7r 2 T4
V(t;c)
m )(T)t;c
AI + 30 A2 t;c - NB + gNF 9()
+ Bt;c 4[ In (t;;) M2 -"6 25]
(2.114)
where
I
m~(T) = m~ + 60(l30AI +47>..2 +75g~)T2
(2.115)
25
4
(2.116)
B = 2567r 2gG
and the zero-temperature radiative correction has been renormalized at mass M as
in section 2.4. In (2.114) the A~ and A~ contributions are always small compared
with the tree terms in (2.114) and have been dropped.
If g~ « AI, A2 we may neglect the zero-temperature radiative correction.
There is then a second-order phase transition with critical temperature Tc given
by
-6Om~
T2....
(2.117)
c - 130A) + 47>..2 + 7Sg~