where ``+ cyclic'' instructs one to add the two cyclic
permutations of legs (2, 3, 4). The scalar planar and non-planar
loop momentum integrals,
and
, are depicted in Fig.
12
. In this expression, all powers of loop momentum have cancelled
from the numerator of each integrand in much the same way as at
one loop, leaving behind only the Feynman propagator
denominators. The explicit values of the two-loop scalar
integrals in terms of polylogarithms may be found in Refs. [127,
133].
The two-loop amplitude (58) has been used by Green, Kwon, and Vanhove [68
] to provide an explicit demonstration of the non-trivial
M-theory duality between
D
=11 supergravity and type II string theory. In this case, the
finite parts of the supergravity amplitudes are important,
particularly the way they depend on the radii of compactified
dimensions.
A remarkable feature of the two-particle cutting
equation (56) is that it can be iterated to
all
loop orders because the tree amplitude (times some scalar
denominators) reappears on the right-hand side. Although this
iteration is insufficient to determine the complete multi-loop
four-point amplitudes, it does provide a wealth of information.
In particular, for planar integrals it leads to the simple
insertion rule depicted in Fig.
13
for obtaining the higher loop contributions from lower loop
ones [19
]. This class includes the contribution in Fig.
4, because it can be assembled entirely from two-particle cuts.
According to the insertion rule, the contribution corresponding
to Fig.
4
is given by loop integrals containing the propagators
corresponding to all the internal lines multiplied by a numerator
factor containing 8 powers of loop momentum. This is to be
contrasted with the 24 powers of loop momentum in the numerator
expected when there are no supersymmetric cancellations. This
reduction in powers of loop momenta leads to improved divergence
properties described in the next subsection.
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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 |