Higher loops

7.2 Higher loops

At two loops, the two-particle cuts are obtained easily by iterating the one-loop calculation, since Eq. (56

) returns a tree amplitude multiplied by some scalar factors. The three-particle cuts are more difficult to obtain, but again one can “recycle” the corresponding cuts used to obtain the two-loop $N = 4$ super-Yang–Mills amplitudes [29 ]. It turns out that the three-particle cuts introduce no other functions than those already detected in the two-particle cuts. After all the cuts are combined into a single function with the correct cuts in all channels, the $N = 8$ supergravity two-loop amplitude [19

] is:

( )6 ℳ2 loop(1,2,3, 4) = κ- s s s M tree(1,2,3,4) 4 2( 12 23 13 4 × s2 ℐ2loop,P(s ,s ) + s2 ℐ2 loop,P (s ,s ) 12 4 12 23 12 4 12 13 +s212ℐ24 loop,NP (s12,s23) + s212ℐ24loop,NP (s12,s13) ) + cyclic , (58 )

where “+ cyclic” instructs one to add the two cyclic permutations of legs (2, 3, 4). The scalar planar and non-planar loop momentum integrals, $2loop,P ℐ4 (s12,s23)$ and $2loop,NP ℐ4 (s12,s23)$ , are depicted in Figure 12

. In this expression, all powers of loop momentum have cancelled from the numerator of each integrand in much the same way as at one loop, leaving behind only the Feynman propagator denominators. The explicit values of the two-loop scalar integrals in terms of polylogarithms may be found in Refs. [127 , 133 ].

Figure 12: The planar and non-planar scalar integrals, appearing in the two-loop $N = 8$ amplitudes. Each internal line represents a scalar propagator.

The two-loop amplitude (58 ) has been used by Green, Kwon, and Vanhove [68 ] to provide an explicit demonstration of the non-trivial M-theory duality between $D = 11$ supergravity and type II string theory. In this case, the finite parts of the supergravity amplitudes are important, particularly the way they depend on the radii of compactified dimensions.

A remarkable feature of the two-particle cutting equation (56 ) is that it can be iterated to all loop orders because the tree amplitude (times some scalar denominators) reappears on the right-hand side. Although this iteration is insufficient to determine the complete multi-loop four-point amplitudes, it does provide a wealth of information. In particular, for planar integrals it leads to the simple insertion rule depicted in Figure 13 for obtaining the higher loop contributions from lower loop ones [19 ]. This class includes the contribution in Figure 4 , because it can be assembled entirely from two-particle cuts. According to the insertion rule, the contribution corresponding to Figure 4 is given by loop integrals containing the propagators corresponding to all the internal lines multiplied by a numerator factor containing 8 powers of loop momentum. This is to be contrasted with the 24 powers of loop momentum in the numerator expected when there are no supersymmetric cancellations. This reduction in powers of loop momenta leads to improved divergence properties described in the next subsection.

Figure 13: Starting from an $l$ -loop planar diagram representing an integral function, an extra line may be added to the inside using this rule. The two lines on the left represent two lines in some $l$ -loop diagram.