Phase transitions in super gravity theories
55
With nG = 3 generations in 5 + 10 and NH sets of Higgs scalars in S, we have
bs = -9+ ~NH
(2.138)
and, for NH = 2,
bs = -8.
(2.139)
It is convenient to choose
M = Mx = 2 x 10 16 GeV
(2.140)
which is the energy scale at which the low-energy (supersymmetric) SU(3) x
SU(2) x U(I) coupling constants reach a common value so that grand unification
may occur. At this scale,
as(Mx) ~ is.
(2.141 )
If we take the criterion for the SU(5) coupling constant to become strong to be
as (T) ~ I, then the corresponding temperature is
T ~ 6.5 x 10- 9 M ~ 10 8 GeV.
(2.142)
It is at this temperature that we expect that either the SU(3) x SU(2) x U(I) or
the SU(4) x U(I) symmetric phase becomes the absolute minimum. Eventually
tunnelling will occurto whichever of these phases is the absolute minimum. If it is
the SU(3) x SU(2) x U(I) symmetric phase, then the universe will continue in this
phase until the coupling constant g4 becomes strong at some lower temperature.
In these globally supersymmetric theories, because the zero-temperature
effective potential is zero when supersymmetry is unbroken, the cosmological
constant is zero in each of the SU(5), SU(4) x U(I) and SU(3) x SU(2) x U(I)
phases until supersymmetry breaking becomes non-negligible (with respect to T)
for temperatures below 10 2 _10 3 GeV.
2.8 Phase transitions in supergravity theories
Up to now we have been discussing theories with global supersymmetry. A
theory with local supersymmetry is necessarily a theory which contains gravity
(supergravity). The reason is that the supersymmetry algebra contains the
generator Pp. of translations and when we allow supersymmetry transformations
that depend on the point in spacetime (local supersymmetry), we have to consider,
among other things, translations that vary from point to point in spacetime. Thus,
local supersymmetry contains general coordinate transformations of spacetime
and so is a theory of gravity.
In phenomenologically acceptable theories, the supersymmetry breaking
scale Ms is large (typically J010-10 11 GeV) where M; is the expectation value
of the auxiliary field of the scalar responsible for supersymmetry breaking. For
example, in theories with F -term supersymmetry breaking, at tree level, a fermion
55
With nG = 3 generations in 5 + 10 and NH sets of Higgs scalars in S, we have
bs = -9+ ~NH
(2.138)
and, for NH = 2,
bs = -8.
(2.139)
It is convenient to choose
M = Mx = 2 x 10 16 GeV
(2.140)
which is the energy scale at which the low-energy (supersymmetric) SU(3) x
SU(2) x U(I) coupling constants reach a common value so that grand unification
may occur. At this scale,
as(Mx) ~ is.
(2.141 )
If we take the criterion for the SU(5) coupling constant to become strong to be
as (T) ~ I, then the corresponding temperature is
T ~ 6.5 x 10- 9 M ~ 10 8 GeV.
(2.142)
It is at this temperature that we expect that either the SU(3) x SU(2) x U(I) or
the SU(4) x U(I) symmetric phase becomes the absolute minimum. Eventually
tunnelling will occurto whichever of these phases is the absolute minimum. If it is
the SU(3) x SU(2) x U(I) symmetric phase, then the universe will continue in this
phase until the coupling constant g4 becomes strong at some lower temperature.
In these globally supersymmetric theories, because the zero-temperature
effective potential is zero when supersymmetry is unbroken, the cosmological
constant is zero in each of the SU(5), SU(4) x U(I) and SU(3) x SU(2) x U(I)
phases until supersymmetry breaking becomes non-negligible (with respect to T)
for temperatures below 10 2 _10 3 GeV.
2.8 Phase transitions in supergravity theories
Up to now we have been discussing theories with global supersymmetry. A
theory with local supersymmetry is necessarily a theory which contains gravity
(supergravity). The reason is that the supersymmetry algebra contains the
generator Pp. of translations and when we allow supersymmetry transformations
that depend on the point in spacetime (local supersymmetry), we have to consider,
among other things, translations that vary from point to point in spacetime. Thus,
local supersymmetry contains general coordinate transformations of spacetime
and so is a theory of gravity.
In phenomenologically acceptable theories, the supersymmetry breaking
scale Ms is large (typically J010-10 11 GeV) where M; is the expectation value
of the auxiliary field of the scalar responsible for supersymmetry breaking. For
example, in theories with F -term supersymmetry breaking, at tree level, a fermion