, where
is a spinor index (for introductions see [178,
166,
169]). As with ordinary symmetries, a theory may be supersymmetric
even though a given state is not supersymmetric; a state which is
annihilated by the supercharges,
, preserves supersymmetry, while states with
are said to spontaneously break SUSY.
Let us begin by considering ``globally supersymmetric'' theories, which are defined in flat spacetime (obviously an inadequate setting in which to discuss the cosmological constant, but we have to start somewhere). Unlike most non-gravitational field theories, in supersymmetry the total energy of a state has an absolute meaning; the Hamiltonian is related to the supercharges in a straightforward way:
where braces represent the anticommutator. Thus, in a
completely supersymmetric state (in which
for all
), the energy vanishes automatically,
[280]. More concretely, in a given supersymmetric theory we can
explicitly calculate the contributions to the energy from vacuum
fluctuations and from the scalar potential
V
. In the case of vacuum fluctuations, contributions from bosons
are exactly canceled by equal and opposite contributions from
fermions when supersymmetry is unbroken. Meanwhile, the
scalar-field potential in supersymmetric theories takes on a
special form; scalar fields
must be complex (to match the degrees of freedom of the
fermions), and the potential is derived from a function called
the superpotential
which is necessarily holomorphic (written in terms of
and not its complex conjugate
). In the simple Wess-Zumino models of spin-0 and spin-1/2
fields, for example, the scalar potential is given by
where
. In such a theory, one can show that SUSY will be unbroken only
for values of
such that
, implying
.
So the vacuum energy of a supersymmetric state in a globally
supersymmetric theory will vanish. This represents rather less
progress than it might appear at first sight, since: 1.)
Supersymmetric states manifest a degeneracy in the mass spectrum
of bosons and fermions, a feature not apparent in the observed
world; and 2.) The above results imply that non-supersymmetric
states have a positive-definite vacuum energy. Indeed, in a state
where SUSY was broken at an energy scale
, we would expect a corresponding vacuum energy
. In the real world, the fact that accelerator experiments have
not discovered superpartners for the known particles of the
Standard Model implies that
is of order
GeV or higher. Thus, we are left with a discrepancy
Comparison of this discrepancy with the naive
discrepancy (54
) is the source of the claim that SUSY can solve the cosmological
constant problem halfway (at least on a log scale).
As mentioned, however, this analysis is strictly valid only in
flat space. In curved spacetime, the global transformations of
ordinary supersymmetry are promoted to the position-dependent
(gauge) transformations of supergravity. In this context the
Hamiltonian and supersymmetry generators play different roles
than in flat spacetime, but it is still possible to express the
vacuum energy in terms of a scalar field potential
. In supergravity
V
depends not only on the superpotential
, but also on a ``Kähler potential''
, and the Kähler metric
constructed from the Kähler potential by
. (The basic role of the Kähler metric is to define the kinetic
term for the scalars, which takes the form
.) The scalar potential is
where
is the Kähler derivative,
(In the presence of gauge fields there will also be
non-negative ``D-terms'', which do not change the present
discussion.) Note that, if we take the canonical Kähler metric
, in the limit
(
) the first term in square brackets reduces to the flat-space
result (56). But with gravity, in addition to the non-negative first term
we find a second term providing a non-positive contribution.
Supersymmetry is unbroken when
; the effective cosmological constant is thus non-positive. We
are therefore free to imagine a scenario in which supersymmetry
is broken in exactly the right way, such that the two terms in
parentheses cancel to fantastic accuracy, but only at the cost of
an unexplained fine-tuning (see for example [63]). At the same time, supergravity is not by itself a
renormalizable quantum theory, and therefore it may not be
reasonable to hope that a solution can be found purely within
this context.
|
The Cosmological Constant
Sean M. Carroll http://www.livingreviews.org/lrr-2001-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
