### 6.2 Arbitrary numbers of legs at one loop

Surprisingly, the above four-point results can be extended to an arbitrary number of external legs. Using
the unitarity methods, the five- and six-point amplitudes with all identical helicity have also
been obtained by direct calculation [22, 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 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 super-Yang–Mills amplitudes was used to
obtain the four-, five-, and six-point 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, 94, 93, 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.