56
Phase transitions in the early universe
has a supersymmebic partner of lower mass than itself as well as one of higher
mass than itself. Since this does not occur in the real world, it is necessary
for there to be significant quantum corrections to avoid this problem, though
not so big that the hierarchy problem is no longer solved. This is the origin
of the large supersymmetry breaking scale. In these circumstances, the effects
of quantum gravity can no longer be neglected. In particular, in the presence of
supersymmetry breaking, scalar particles acquire masses of order M; / m p (where
mp ~ 1.2 x }OI9 GeV is the Planck mass) which are oforcier 102 GeV.
Once gravitational effects are important, we should allow not only that the
superpotential may contain non-renormalizable terms but also that there may be
non-renormalizable kinetic terms. Thus, for example, the scalar field kinetic terms
take the form
o2K
(2.143)
OlPiOlPj 0l£tPiOl"lPj
where K(lPi, lPi> is referred to as the Kiihler potential. It turns out [13] that the
complete supergravity Lagrangian can be expressed in terms of
G = K +lnlWl 2
(2.144)
apart from couplings to gauge fields, which involve the gauge kinetic function. It
will often be convenient to work in units where the reduced Planck mass Mp = I,
where Mp is defined by
Mp2 == 87rGN
(2.145)
where G N is Newton's constant, so that
Mp ~ 2.44 x 10 18 GeV.
(2.146)
In these units, the zero-temperature effective potential takes the form
V = eG(Gi(G-I)~Gj - 3)
(2.147)
where the scalar fields have been written as lPi. their adjoints as lP i • and derivatives
ofG as
oG
Gi == oG
(2.148)
Gi == alPi•
olPi
and
a 2 G
i
__ •
(2.149)
G j == alPi olPj
The inverse (G-I)~ obeys
(G-I)~G~ = 81.
(2.150)
In particular, in the case of a single gauge-singlet chiral superfield ~ with minimal
kinetic terms arising from
G = lP·lP + In IWl 2
(2.151 )