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
Theory
Zvi Bern http://www.livingreviews.org/lrr-2002-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |