The two-loop five-point amplitude in $\mathcal N=8$ supergravity

TL;DR

This work delivers the symbol of the two-loop five-point amplitude in ${ m N}=8$ SUGRA by combining a $d$-dimensional unitarity-based construction of rational prefactors with a numerical IBP-driven reconstruction of master-integral contributions. The key advance is the identification of a 45-dimensional basis of rational coefficients, arising from both 4D leading singularities and genuine $d$-dimensional cuts, which, together with a canonical-differential-equation analysis of a pure master-integral basis, yields a uniformly transcendental, infrared-consistent result. The finite, infrared-subtracted remainder is shown to depend on 40 weight-4 functions, with a large overlap with the ${ m N}=4$ SYM function space; the analysis highlights structural parallels with gauge theory and points to potential cross-fertilization with QCD computations. Soft and collinear limits verify the expected universal factorization in gravity, and the work sets the stage for exploring BCJ-duality imprints in full amplitudes and extending dimensional-cut techniques to other theories.

Abstract

We compute the symbol of the two-loop five-point amplitude in $\mathcal N=8$ supergravity. We write an ansatz for the amplitude whose rational prefactors are based on not only 4-dimensional leading singularities, but also $d$-dimensional ones, as the former are insufficient. Our novel $d$-dimensional unitarity-based approach to the systematic construction of an amplitude's rational structures is likely to have broader applications, for example to analogous QCD calculations. We fix parameters in the ansatz by performing numerical integration-by-parts reduction of the known integrand. We find that the two-loop five-point $\mathcal N=8$ supergravity amplitude is uniformly transcendental. We then verify the soft and collinear limits of the amplitude. There is considerable similarity with the corresponding amplitude for $\mathcal N=4$ super-Yang-Mills theory: all the rational prefactors are double copies of the Yang-Mills ones and the transcendental functions overlap to a large degree. As a byproduct, we find new relations between color-ordered loop amplitudes in $\mathcal N=4$ super-Yang-Mills theory.

Figures (4)

• Figure 1: Diagram topologies entering the local representation of the two-loop five-point integrand of $\mathcal{N}=8$ supergravity Carrasco:2011mn. Each diagram has an associated kinematic numerator which we give in the main text.
• Figure 2: The double-box diagram.
• Figure 3: The six parity-odd $(6-2\epsilon)$-dimensional master integrals and their normalization factors. The integrals have no numerators. A red dot indicates that the propagator is "doubled" i.e. raised to a squared power.
• Figure 4: A spanning set of cuts for performing IBP reduction for the nonplanar double pentagon diagram. A cut propagator is indicated by a red line. There are 11 cuts in total, from applying diagram symmetries to the 4 representative cuts shown here.