### 7.1 One-loop cut construction

The maximal supergravity amplitudes can be obtained by applying the KLT equations to
express them in terms of maximally supersymmetric gauge theory amplitudes. For
supergravity, each of the states of the multiplet factorizes into a tensor product of
super-Yang–Mills states, as illustrated in Eq. (9). Applying the KLT equation (10) to the product of tree
amplitudes appearing in the channel two-particle cuts yields:
where the sum on the left-hand side runs over all 256 states in the supergravity multiplet. On the
right-hand side the two sums run over the 16 states (ignoring color degrees of freedom) of the
super-Yang–Mills multiplet: a gluon, four Weyl fermions and six real scalars.
The super-Yang–Mills tree amplitudes turn out to have a particularly simple sewing
formula [29],

which holds in any dimension (though some care is required to maintain the total number of physical states
at their four-dimensional values so as to preserve the supersymmetric cancellations). The simplicity of this
result is due to the high degree of supersymmetry.
Using the gauge theory result (55), it is a simple matter to evaluate Eq. (54). This yields:

The sewing equations for the and kinematic channels are similar to that of the
channel.
Applying Eq. (56) at one loop to each of the three kinematic channels yields the one-loop four graviton
amplitude of supergravity,

in agreement with previous results [69]. The gravitational coupling has been reinserted into this
expression. The scalar integrals are defined in Eq. (51), inserting . This is a standard
integral appearing in massless field theories; the explicit value of this integral may be found
in many articles, including Refs. [69, 27]. This result actually holds for any of the states of
supergravity, not just external gravitons. It is also completely equivalent to the result
one obtains with covariant Feynman diagrams including Fadeev–Popov [59] ghosts and using
regularization by dimensional reduction [122]. The simplicity of this result is due to the high degree of
supersymmetry. A generic one-loop four-point gravity amplitude can have up to eight powers of loop
momenta in the numerator of the integrand; the supersymmetry cancellations have reduced it to no
powers.