58
Phase transitions in the early universe
the case that the kinetic terms are non-minimal, we first construct fields with
i
)
)
s
)
)
)
)
r
s
r
f
r
)
t
l
canonical kinetic terms by field redefinition. This means that. for scalar fields tP
(and their fermionic superpartners), we have to write
tPi = (G-I/l>{(tPj)N
(2.160
and for the gauginos
Aa = (Re fab)-1/2(Ab)N
(2.161
where (tPj)N and (Ab)N are the normalized fields. The relevant mass matrices
are obtained from the standard supergravity Lagrangian. The outcome [15-18] i
particularly simple in the case of minimal kinetic terms. Then
-T
1r2T4 (
7)
VI = - - - NB+-NF
90
8
T2 [3
I ]
+ l2eG 2(A + B) + (C + N)(C - 2) + 2C2 + C - I (2.162
where
A = dGijGj + GiGijGj
(2.163
B = Gi)d)
(2.164
C=GiG i
(2.165
and N is the number of chiral superfields. In practice, N is often large. Fo
example, if the matter field content is that of the minimal SU(5) GUT, there are
nG = 3 generations in the 5 + 10 representation contributing 45 to N, two copie
of 5 or 5 for electroweak Higgs contributing \0 to N and one copy of 24 fo
the grand unified Higgs scalars contributing 24 to N, leading to a total value o
N=79.
Provided that all of the couplings in the superpotential are of the same orde
of magnitude, in units where the reduced Planck mass is I, we can then take the
large-N limit to obtain
-T
1r2T4 (
7) T2 G .
VI = - - - NB + -NF +N-e (GiG' -2)
(2.166
90
8
12
.
Provided that the changes in V~ generated by non-minimal kinetic terms do no
introduce extra factors of N (which is true for most choices of G), this is still the
large-N limit of vf in that case. For any particular choice of KIDder potential and
superpotential and, therefore, of G, the discussion of phase transitions proceeds as
before with the modified finite-temperature corrections to the effective potentia
of (2.162).