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].
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 |