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.
Highlights
The precision goals for current and future collider experiments
At the level of amplitudes, rather than integrands, whilst considerable progress has been made in the planar sector of N = 4 SYM, much less is known beyond the planar limit
The complete set of leading-color five-point twoloop planar amplitudes in QCD is known numerically [48, 68,69,70] and the two-loop five-gluon scattering amplitudes in pure Yang-Mills are known analytically [59, 71, 72]. These methods have led to the first analytic results for the symbol of the two-loop five-point N = 4 SYM amplitude including nonplanar contributions [65, 73]. This amplitude is simpler to compute than the one we study in this paper because its integrand only involves numerators with one power of loop momentum, while in N = 8 SUGRA the numerators have two powers of loop momentum [24]
Summary
All known amplitudes in N = 4 SYM and N = 8 SUGRA share the common feature of being functions of uniform transcendental (UT) weight [31, 36, 37, 79, 85] Whether this property persists at higher numbers of loops or legs is an outstanding open question which the present work touches on. There are known pieces in the integrand [85] that have non-logarithmic poles at infinity, which are expected to cause a transcendentality drop. Whether such contributions cancel in the final amplitudes — similar in spirit to enhanced cancellations of UV divergences These rational functions are (linear combinations of) the leading singularities we shall be discussing .
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