7.3 Divergence properties of N = 8 supergravity

Since the two-loop N = 8 supergravity amplitude (58View Equation) has been expressed in terms of scalar integrals, it is straightforward to extract the divergence properties. The scalar integrals diverge only for dimension D ≥ 7; hence the two-loop N = 8 amplitude is manifestly finite in D = 5 and 6, contrary to earlier expectations based on superspace power counting [81Jump To The Next Citation Point]. 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 [129Jump To The Next Citation Point]. 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 (58View Equation) is

two−loop,D=7− 2𝜖|| 1 π 2 2 2 (κ )6 tree ℳ 4 |div.= 2𝜖(4π-)73-(s12 + s23 + s13) 2- × s12s23s13 M 4 , (59 )
which clearly diverges when the dimensional regularization parameter 𝜖 → 0.

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 [63738]. 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 [78]. 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 [67116Jump To The Next Citation Point68].

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 (58View Equation) 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. [4041]. 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. [15].

Using the insertion rule of Figure 13View Image, and counting the powers of loop momenta in these contributions leads to the simple finiteness condition

--10-- l < D − 2 (60 )
(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 [81Jump To The Next Citation Point], 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 [19Jump To The Next Citation Point]. Moreover, a recent improved superspace power count [129Jump To The Next Citation Point], taking into account a higher-dimensional gauge symmetry, is in agreement with the finiteness formula (60View Equation). 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 [19Jump To The Next Citation Point], contrary to earlier expectations from superspace power counting [81].

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