### 7.2 Higher loops

At two loops, the two-particle cuts are obtained easily by iterating the one-loop calculation, since
Eq. (56) returns a tree amplitude multiplied by some scalar factors. The three-particle cuts are more
difficult to obtain, but again one can “recycle” the corresponding cuts used to obtain the two-loop
super-Yang–Mills amplitudes [29]. It turns out that the three-particle cuts introduce no other functions
than those already detected in the two-particle cuts. After all the cuts are combined into a single function
with the correct cuts in all channels, the supergravity two-loop amplitude [19] is:
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
Figure 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 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 Figure 13 for obtaining the higher loop contributions from lower loop ones [19]. This class
includes the contribution in Figure 4, because it can be assembled entirely from two-particle cuts.
According to the insertion rule, the contribution corresponding to Figure 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.