) has been expressed in terms of scalar integrals, it is
straightforward to extract the divergence properties. The scalar
integrals diverge only for dimension
; hence the two-loop
N
=8 amplitude is manifestly finite in
D
=5 and 6, contrary to earlier expectations based on superspace
power counting [80
]. The discrepancy between the above explicit results and the
earlier superspace power counting arguments may be understood in
terms of an unaccounted higher-dimensional gauge symmetry [128]. Once this symmetry is accounted for, superspace power counting
gives the same degree of divergence as the explicit calculation.
The cutting methods provide much more than just an indication
of divergence; one can extract the explicit numerical
coefficients of the divergences. For example, near
D
=7 the divergence of the amplitude (58) is
which clearly diverges when the dimensional regularization
parameter
.
In all cases the linearized divergences take the form of
derivatives acting on a particular contraction of Riemann
tensors, which in four dimensions is equivalent to the square of
the Bel-Robinson tensor [6,
37,
38]. This operator appears in the first set of corrections to the
N
=8 supergravity Lagrangian, in the inverse string-tension
expansion of the effective field theory for the type II
superstring [77]. Therefore, it has a completion into an
N
=8 supersymmetric multiplet of operators, even at the non-linear
level. It also appears in the M-theory one-loop and two-loop
effective actions [67,
116,
68].
Interestingly, the manifest
D
-independence of the cutting algebra allows the calculation to be
extended to
D
=11, even though there is no corresponding
D
=11 super-Yang-Mills theory. The result (58) then explicitly demonstrates that
N
=1,
D
=11 supergravity diverges. In dimensional regularization there
are no one-loop divergences so the first potential divergence is
at two loops. (In a momentum cutoff scheme the divergences
actually begin at one loop [116].) Further work on the structure of the
D
=11 two-loop divergences in dimensional regularization has been
carried out in Ref. [40,
41]. The explicit form of the linearized
N
=1,
D
=11 counterterm expressed as derivatives acting on Riemann
tensors along with a more general discussion of supergravity
divergences may be found in Ref. [17].
Using the insertion rule of Fig. 13, and counting the powers of loop momenta in these contributions leads to the simple finiteness condition
(with
l
>1), where
l
is the number of loops. This formula indicates that
N
=8 supergravity is finite in some other cases where the previous
superspace bounds suggest divergences [80],
e.g.
D
=4,
l
=3: The first
D
=4 counterterm detected via the two-particle cuts of four-point
amplitudes occurs at five, not three loops. Further evidence that
the finiteness formula is correct stems from the maximally
helicity violating contributions to
m
-particle cuts, in which the same supersymmetry cancellations
occur as for the two-particle cuts [19]. Moreover, a recent improved superspace power count [128], taking into account a higher-dimensional gauge symmetry, is in
agreement with the finiteness formula (60). Further work would be required to prove that other
contributions do not alter the two-particle cut power counting. A
related open question is whether one can prove that the five-loop
D
=4 divergence encountered in the two-particle cuts does not
somehow cancel against other contributions [32] because of some additional symmetry. It would also be
interesting to explicitly demonstrate the non-existence of
divergences after including all contributions to the three-loop
amplitude. In any case, the explicit calculations using cutting
methods do establish that at two loops maximally supersymmetric
supergravity does not diverge in
D
=5 [19], contrary to earlier expectations from superspace power
counting [80].
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Perturbative Quantum Gravity and its Relation to Gauge
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Zvi Bern http://www.livingreviews.org/lrr-2002-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
