). 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 N =8 supergravity multiplet. On the right-hand side the two sums run over the 16 states (ignoring color degrees of freedom) of the N =4 super-Yang-Mills multiplet: a gluon, four Weyl fermions and six real scalars.
The
N
=4 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
N
=8 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
N
=8 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.
|
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 |
