The Polonyi problem
239
If the field ~ starts (because of quantum fluctuations) at some value
of ~ which differs from the minimum, then the energy density stored in
the expectation value of ~ is of order il4. The size of il is related to
the size of supersymmetry breaking effects. In supergravity theories with
supersymmetry breaking in a hidden sector, the size of supersymmetry breaking
effects transmitted gravitationally to the observable sector is on the scale of the
gravitino mass m3/2 and. for supersymmetry to solve the hierarchy problem, m3/2
should be about 100 GeV to 10 TeV. In the Polonyi model. the gravitino mass is
il 2 ". 2
m3/2 = _e(,.13-1) /2.
(8.93)
Mp
Thus, for m3/2 in the range 100 GeV to 10 TeV, we have
1010 GeV ~ il ~ 1011 GeV.
(8.94)
(For a discussion of the gravitino mass in supergravity theories with hidden-sector
supersymmetry breaking see, for example, [12].)
For the discussion of the entropy increase of the universe when the Polonyi
field vacuum energy decays, we shall need to know the expectation value of
the Polonyi field. In the context of cosmological inflation, we should determine
this expectation value by minimizing the total effective potential of the Polonyi
field and the inflaton. (There may also be effects of quantum fluctuations.) The
superpotential of the Polonyi field ~ is as in (8.87) and. for definiteness, we may
take the superpotential for the inflaton field tP to be
W(tP) = J,lh.2(tP - a)2
(8.95)
as in (8.9). Thus, the total superpotential is
Wtot(q,,~) = W(~) + W(tP).
(8.96)
We also assume minimal kinetic terms so that
G = ~*~ + tP*tP + In IWtotl 2 •
(8.97)
Then the effective potential can be calculated from (2.141) in units where the
reduced Planck mass Mp = I. Taking ~ to be real, working to quadratic order in
~ (when ~ « 1 in the same units), and remembering that tP ~ 0 during slow roll,
the minimum of the effective potential may be estimated to be (exercise 5)
-
2JL 2 ilJ..2lT 2
(8.98)
q, ~ JL4J..~ - 4f3J.1.2ili: 2 d-·
The values of the parameters of the Polonyi model are given by (8.89) and (8.94),
and the parameters of the superstring model of inflation that we are employing by
(8.13) and (8.33). Using these values, ~ may be estimated to be
~ ~ 3(10- 3 -10- 4 ).
(8.99)
239
If the field ~ starts (because of quantum fluctuations) at some value
of ~ which differs from the minimum, then the energy density stored in
the expectation value of ~ is of order il4. The size of il is related to
the size of supersymmetry breaking effects. In supergravity theories with
supersymmetry breaking in a hidden sector, the size of supersymmetry breaking
effects transmitted gravitationally to the observable sector is on the scale of the
gravitino mass m3/2 and. for supersymmetry to solve the hierarchy problem, m3/2
should be about 100 GeV to 10 TeV. In the Polonyi model. the gravitino mass is
il 2 ". 2
m3/2 = _e(,.13-1) /2.
(8.93)
Mp
Thus, for m3/2 in the range 100 GeV to 10 TeV, we have
1010 GeV ~ il ~ 1011 GeV.
(8.94)
(For a discussion of the gravitino mass in supergravity theories with hidden-sector
supersymmetry breaking see, for example, [12].)
For the discussion of the entropy increase of the universe when the Polonyi
field vacuum energy decays, we shall need to know the expectation value of
the Polonyi field. In the context of cosmological inflation, we should determine
this expectation value by minimizing the total effective potential of the Polonyi
field and the inflaton. (There may also be effects of quantum fluctuations.) The
superpotential of the Polonyi field ~ is as in (8.87) and. for definiteness, we may
take the superpotential for the inflaton field tP to be
W(tP) = J,lh.2(tP - a)2
(8.95)
as in (8.9). Thus, the total superpotential is
Wtot(q,,~) = W(~) + W(tP).
(8.96)
We also assume minimal kinetic terms so that
G = ~*~ + tP*tP + In IWtotl 2 •
(8.97)
Then the effective potential can be calculated from (2.141) in units where the
reduced Planck mass Mp = I. Taking ~ to be real, working to quadratic order in
~ (when ~ « 1 in the same units), and remembering that tP ~ 0 during slow roll,
the minimum of the effective potential may be estimated to be (exercise 5)
-
2JL 2 ilJ..2lT 2
(8.98)
q, ~ JL4J..~ - 4f3J.1.2ili: 2 d-·
The values of the parameters of the Polonyi model are given by (8.89) and (8.94),
and the parameters of the superstring model of inflation that we are employing by
(8.13) and (8.33). Using these values, ~ may be estimated to be
~ ~ 3(10- 3 -10- 4 ).
(8.99)