Four-graviton scattering to three loops in $$ \mathcal{N}=8 $$ supergravity

Abstract

We compute the three-loop scattering amplitude of four gravitons in N = 8 supergravity. Our results are analytic formulae for a Laurent expansion of the amplitude in the regulator of dimensional regularisation. The coefficients of this series are closed formulae in terms of well-established harmonic poly-logarithms. Our results display a remarkable degree of simplicity and represent an important stepping stone in the exploration of the structure of scattering amplitudes. In particular, we observe that to this loop order the four graviton amplitude is given by uniform weight 2L functions, where L is the loop order.

A tale of two exponentiations in N=8 supergravity

“…a quantity that diverges as 2 ǫ for small ǫ. This expression is valid for general values of ǫ up to the subleading level in the Regge limit and we checked that in this regime it reproduces the data of [52] where the one-loop result is written explicitly up to O(ǫ 4 ).…”

“…At this order the amplitudes depend on the details of the theory and we focused on the case of N = 8 supergravity. 16 By using the explicit results of [52] we compared the two exponentiations at all loops for the first four terms in the ǫ → 0 expansion. As discussed in section 5, there is an impressive agreement between the eikonal prediction (2.15) and the explicit results of [52] that satisfy perfectly the IR exponentation in momentum space (2.4).…”

“…16 By using the explicit results of [52] we compared the two exponentiations at all loops for the first four terms in the ǫ → 0 expansion. As discussed in section 5, there is an impressive agreement between the eikonal prediction (2.15) and the explicit results of [52] that satisfy perfectly the IR exponentation in momentum space (2.4). However there is a mismatch for one term appearing at the lowest power of 1/ǫ and the lowest power of log(q 2 ) accessible with the current data.…”

Constructing d-log integrands and computing master integrals for three-loop four-particle scattering

“…Analytic continuation into another scattering region may be performed for example by following the steps discussed in ref. [57,58]. Similarly, permutations of external legs of our master integrals can be obtained using the methods detailed in refs.…”

“…Similarly, permutations of external legs of our master integrals can be obtained using the methods detailed in refs. [57,58]. We checked that the permutations of our master integrals satisfy the permuted systems of differential equations.…”