where the superscript
s
indicates that the cut lines are scalars. The
h
subscripts on legs
indicate that these are gravitons, while the ``+'' superscripts
indicate that they are of plus helicity. From the KLT
expressions (10
) the gravity tree amplitudes appearing in the cuts may be
replaced with products of gauge theory amplitudes. The required
gauge theory tree amplitudes, with two external scalar legs and
two gluons, may be obtained using color-ordered Feynman diagrams
and are
The external gluon momenta are four-dimensional, but the
scalar momenta
and
are
D
-dimensional since they will form the loop momenta. In general,
loop momenta will have a non-vanishing
-dimensional component
, with
. The factors of
appearing in the numerators of these tree amplitudes causes them
to vanish as the scalar momenta are taken to be four-dimensional,
though they are non-vanishing away from four dimensions. For
simplicity, overall phases have been removed from the amplitudes.
After inserting these gauge theory amplitudes in the KLT
relation (10
), one of the propagators cancels, leaving
For this cut, one then obtains a sum of box integrals that can be expressed as
By symmetry, since the helicities of all the external gravitons are identical, the other two cuts also give the same combinations of box integrals, but with the legs permuted.
The three cuts can then be combined into a single function that has the correct cuts in all channels yielding
and where
is the box integral depicted in Fig.
11
with the external legs arranged in the order 1234. In Eq. (50)
is
. The two other integrals that appear correspond to the two other
distinct orderings of the four external legs. The overall factor
of 2 in Eq. (50
) is a combinatoric factor due to taking the scalars to be
complex with two physical states.
Since the factor of
is of
, the only non-vanishing contributions come where the
from the
interferes with a divergence in the loop integral. These
divergent contributions are relatively simple to obtain. After
extracting this contribution from the integral, the final
D
=4 result for a complex scalar loop, after reinserting the
gravitational coupling, is
in agreement with a calculation done by a different method relying directly on string theory [55]. (As for the previous expressions, the overall phase has been suppressed.)
This result generalizes very simply to the case of any particles in the loop. For any theory of gravity, with an arbitrary matter content one finds:
where
is the number of physical bosonic states circulating in the loop
minus the number of fermionic states. The simplest way to
demonstrate this is by making use of supersymmetry Ward
identities [71,
104,
20
], which provide a set of simple linear relations between the
various contributions showing that they must be proportional to
each other.
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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 |