,
23]. Then by demanding that the amplitudes have the properties
described in Section
3.4
for momenta becoming either soft [139,
10] or collinear [22], an ansatz for the one-loop maximally helicity-violating
amplitudes for an arbitrary number of external legs has also been
obtained. These amplitudes were constructed from a set of
building blocks called ``half-soft-function'', which have
``half'' of the proper behavior as gravitons become soft. The
details of this construction and the explicit forms of the
amplitudes may be found in Refs. [22,
23].
The all-plus helicity amplitudes turn out to be very closely
related to the infinite sequence of one-loop maximally
helicity-violating amplitudes in
N
=8 supergravity. The two sequences are related by a curious
``dimension shifting formula.'' In Ref. [23], a known dimension shifting formula [18] between identical helicity QCD and
N
=4 super-Yang-Mills amplitudes was used to obtain the four-,
five-, and six-point
N
=8 amplitudes from the identical helicity gravity amplitudes
using the KLT relations in the unitarity cuts. Armed with these
explicit results, the soft and collinear properties were then
used to obtain an ansatz valid for an arbitrary number of
external legs [23]. This provides a rather non-trivial illustration of how the KLT
relations can be used to identify properties of gravity
amplitudes using known properties of gauge theory amplitudes.
Interestingly, the all-plus helicity amplitudes are also
connected to self-dual gravity [108,
52,
109] and self-dual Yang-Mills [143,
53,
93,
92,
4,
30,
33],
i.e.
gravity and gauge theory restricted to self-dual configurations
of the respective field strengths,
and
, with
. This connection is simple to see at the linearized (free field
theory) level since a superposition of plane waves of identical
helicity satisfies the self-duality condition. The self-dual
currents and amplitudes have been studied at tree and one-loop
levels [53,
4,
30,
33]. In particular, Chalmers and Siegel [33] have presented self-dual actions for gauge theory (and
gravity), which reproduce the all-plus helicity scattering
amplitudes at both tree and one-loop levels.
The ability to obtain exact expressions for gravity loop amplitudes demonstrates the utility of this approach for investigating quantum properties of gravity theories. The next section describes how this can be used to study high energy divergence properties in quantum gravity.
|
Perturbative Quantum Gravity and its Relation to Gauge
Theory
Zvi Bern http://www.livingreviews.org/lrr-2002-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |